Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.8s

Alternatives: 12

Speedup: 1.0×

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, 1.0× 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]]

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs((((eh * cos(t)) * sin(atan(t_1))) + ((ew * sin(t)) / hypot(1.0, t_1))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. 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. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (ew / (hypot(1.0, (eh / (ew * tan(t)))) / sin(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[(ew / N[(N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 60.7%

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

   2. pow2 60.7%

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

   3. associate-*l* 60.7%

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

   4. cos-atan 77.4%

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

   5. un-div-inv 77.4%

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

   6. hypot-1-def 74.7%

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

4. Applied egg-rr 74.7%

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

   1. unpow2 74.7%

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

   2. add-sqr-sqrt 99.8%

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

   3. clear-num 99.7%

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

   4. un-div-inv 99.7%

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

   5. associate-/l/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 98.9%

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

Alternative 6: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7 \cdot 10^{-84} \lor \neg \left(ew \leq 1.8 \cdot 10^{-93}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7e-84) (not (<= ew 1.8e-93)))
   (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ (/ eh ew) (tan t)))))))
   (fabs (* (* eh (cos t)) (sin (atan (/ (/ eh (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7e-84) || !(ew <= 1.8e-93)) {
		tmp = fabs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs(((eh * cos(t)) * sin(atan(((eh / tan(t)) / ew)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-7d-84)) .or. (.not. (ew <= 1.8d-93))) then
        tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))))
    else
        tmp = abs(((eh * cos(t)) * sin(atan(((eh / tan(t)) / ew)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7e-84) || !(ew <= 1.8e-93)) {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / Math.tan(t)) / ew)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -7e-84) or not (ew <= 1.8e-93):
		tmp = math.fabs(((ew * math.sin(t)) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = math.fabs(((eh * math.cos(t)) * math.sin(math.atan(((eh / math.tan(t)) / ew)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7e-84) || !(ew <= 1.8e-93))
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -7e-84) || ~((ew <= 1.8e-93)))
		tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	else
		tmp = abs(((eh * cos(t)) * sin(atan(((eh / tan(t)) / ew)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7e-84], N[Not[LessEqual[ew, 1.8e-93]], $MachinePrecision]], 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], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.0000000000000002e-84 or 1.8000000000000001e-93 < ew

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 56.0%

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

      2. pow2 56.0%

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

      3. associate-*l* 56.0%

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

      4. cos-atan 69.3%

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

      5. un-div-inv 69.3%

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

      6. hypot-1-def 64.9%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in ew around inf 55.8%

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

      1. unpow2 55.8%

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

      2. add-sqr-sqrt 99.1%

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

      3. *-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in t around 0 92.4%

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

   if -7.0000000000000002e-84 < ew < 1.8000000000000001e-93

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 68.2%

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

      2. pow2 68.2%

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

      3. associate-*l* 68.2%

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

      4. cos-atan 90.6%

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

      5. un-div-inv 90.6%

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

      6. hypot-1-def 90.6%

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in ew around 0 93.7%

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

      1. associate-*r* 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-/r* 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7 \cdot 10^{-84} \lor \neg \left(ew \leq 1.8 \cdot 10^{-93}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.37 \lor \neg \left(t \leq 0.0032\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.37) (not (<= t 0.0032)))
   (fabs (* (* eh (cos t)) (sin (atan (/ (/ eh (tan t)) ew)))))
   (fabs (+ (* ew t) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.37) || !(t <= 0.0032)) {
		tmp = fabs(((eh * cos(t)) * sin(atan(((eh / tan(t)) / ew)))));
	} else {
		tmp = fabs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.37d0)) .or. (.not. (t <= 0.0032d0))) then
        tmp = abs(((eh * cos(t)) * sin(atan(((eh / tan(t)) / ew)))))
    else
        tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.37) || !(t <= 0.0032)) {
		tmp = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / Math.tan(t)) / ew)))));
	} else {
		tmp = Math.abs(((ew * t) + (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.37) or not (t <= 0.0032):
		tmp = math.fabs(((eh * math.cos(t)) * math.sin(math.atan(((eh / math.tan(t)) / ew)))))
	else:
		tmp = math.fabs(((ew * t) + (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.37) || !(t <= 0.0032))
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / tan(t)) / ew)))));
	else
		tmp = abs(Float64(Float64(ew * t) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.37) || ~((t <= 0.0032)))
		tmp = abs(((eh * cos(t)) * sin(atan(((eh / tan(t)) / ew)))));
	else
		tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.37], N[Not[LessEqual[t, 0.0032]], $MachinePrecision]], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * t), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.37 or 0.00320000000000000015 < t

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 57.2%

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

      2. pow2 57.2%

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

      3. associate-*l* 57.2%

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

      4. cos-atan 77.5%

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

      5. un-div-inv 77.5%

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

      6. hypot-1-def 73.3%

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

   4. Applied egg-rr 73.3%

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

   5. Taylor expanded in ew around 0 58.1%

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

      1. associate-*r* 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-/r* 58.1%

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

   7. Simplified 58.1%

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

   if -0.37 < t < 0.00320000000000000015

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 63.6%

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

      2. pow2 63.6%

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

      3. associate-*l* 63.6%

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

      4. cos-atan 77.3%

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

      5. un-div-inv 77.3%

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

      6. hypot-1-def 75.9%

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

   4. Applied egg-rr 75.9%

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

   5. Taylor expanded in ew around inf 63.6%

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

   6. Taylor expanded in t around 0 97.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.37 \lor \neg \left(t \leq 0.0032\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.65 \cdot 10^{-171} \lor \neg \left(eh \leq 3.8 \cdot 10^{-178}\right):\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{{\left(ew \cdot \sin t\right)}^{3}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.65e-171) (not (<= eh 3.8e-178)))
   (fabs (* eh (sin (atan (/ eh (* ew (tan t)))))))
   (fabs (cbrt (pow (* ew (sin t)) 3.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.65e-171) || !(eh <= 3.8e-178)) {
		tmp = fabs((eh * sin(atan((eh / (ew * tan(t)))))));
	} else {
		tmp = fabs(cbrt(pow((ew * sin(t)), 3.0)));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.65e-171) || !(eh <= 3.8e-178)) {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	} else {
		tmp = Math.abs(Math.cbrt(Math.pow((ew * Math.sin(t)), 3.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.65e-171) || !(eh <= 3.8e-178))
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	else
		tmp = abs(cbrt((Float64(ew * sin(t)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.65e-171], N[Not[LessEqual[eh, 3.8e-178]], $MachinePrecision]], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Power[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.6500000000000001e-171 or 3.80000000000000015e-178 < eh

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 60.8%

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

      2. pow2 60.8%

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

      3. associate-*l* 60.8%

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

      4. cos-atan 81.4%

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

      5. un-div-inv 81.4%

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

      6. hypot-1-def 78.1%

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

   4. Applied egg-rr 78.1%

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

   5. Taylor expanded in t around 0 48.3%

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

   if -1.6500000000000001e-171 < eh < 3.80000000000000015e-178

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 60.0%

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

      2. pow2 60.0%

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

      3. associate-*l* 60.0%

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

      4. cos-atan 60.0%

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

      5. un-div-inv 60.0%

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

      6. hypot-1-def 60.0%

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

   4. Applied egg-rr 60.0%

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

   5. Taylor expanded in ew around inf 60.0%

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

      1. add-cbrt-cube 30.1%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot \left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \cdot \left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}}\right| \]

      2. pow3 30.1%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{3}}}\right| \]

      3. +-commutative 30.1%

         \[\leadsto \left|\sqrt[3]{{\color{blue}{\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + {\left(\sqrt{ew \cdot \sin t}\right)}^{2}\right)}}^{3}}\right| \]

      4. fma-define 30.1%

         \[\leadsto \left|\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(eh \cdot \cos t, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), {\left(\sqrt{ew \cdot \sin t}\right)}^{2}\right)\right)}}^{3}}\right| \]

      5. unpow2 30.1%

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

      6. add-sqr-sqrt 50.2%

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

   7. Applied egg-rr 50.2%

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

   8. Taylor expanded in eh around 0 32.3%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3} \cdot {\sin t}^{3}}}\right| \]
   9. Step-by-step derivation

      1. cube-prod 50.2%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \sin t\right)}^{3}}}\right| \]

   10. Simplified 50.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.65 \cdot 10^{-171} \lor \neg \left(eh \leq 3.8 \cdot 10^{-178}\right):\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{{\left(ew \cdot \sin t\right)}^{3}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{{\left(ew \cdot \sin t\right)}^{3}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 1.05e-6)
   (fabs (+ (* ew t) (* eh (sin (atan (/ eh (* ew (tan t))))))))
   (fabs (cbrt (pow (* ew (sin t)) 3.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 1.05e-6) {
		tmp = fabs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = fabs(cbrt(pow((ew * sin(t)), 3.0)));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 1.05e-6) {
		tmp = Math.abs(((ew * t) + (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	} else {
		tmp = Math.abs(Math.cbrt(Math.pow((ew * Math.sin(t)), 3.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 1.05e-6)
		tmp = abs(Float64(Float64(ew * t) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = abs(cbrt((Float64(ew * sin(t)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 1.05e-6], N[Abs[N[(N[(ew * t), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Power[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.0499999999999999e-6

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 59.9%

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

      2. pow2 59.9%

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

      3. associate-*l* 59.9%

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

      4. cos-atan 78.0%

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

      5. un-div-inv 78.0%

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

      6. hypot-1-def 75.4%

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

   4. Applied egg-rr 75.4%

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

   5. Taylor expanded in ew around inf 59.4%

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

   6. Taylor expanded in t around 0 72.7%

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

   if 1.0499999999999999e-6 < t

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 63.1%

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

      2. pow2 63.1%

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

      3. associate-*l* 63.1%

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

      4. cos-atan 75.7%

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

      5. un-div-inv 75.7%

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

      6. hypot-1-def 72.6%

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in ew around inf 63.1%

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

      1. add-cbrt-cube 30.0%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot \left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \cdot \left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}}\right| \]

      2. pow3 30.0%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{3}}}\right| \]

      3. +-commutative 30.0%

         \[\leadsto \left|\sqrt[3]{{\color{blue}{\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + {\left(\sqrt{ew \cdot \sin t}\right)}^{2}\right)}}^{3}}\right| \]

      4. fma-define 30.0%

         \[\leadsto \left|\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(eh \cdot \cos t, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), {\left(\sqrt{ew \cdot \sin t}\right)}^{2}\right)\right)}}^{3}}\right| \]

      5. unpow2 30.0%

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

      6. add-sqr-sqrt 41.3%

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

   7. Applied egg-rr 41.3%

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

   8. Taylor expanded in eh around 0 25.2%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3} \cdot {\sin t}^{3}}}\right| \]
   9. Step-by-step derivation

      1. cube-prod 25.2%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \sin t\right)}^{3}}}\right| \]

   10. Simplified 25.2%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{{\left(ew \cdot \sin t\right)}^{3}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -9.4 \cdot 10^{-20} \lor \neg \left(eh \leq 7.4 \cdot 10^{-178}\right):\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{{\left(ew \cdot \sin t\right)}^{3}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -9.4e-20) (not (<= eh 7.4e-178)))
   (fabs (* eh (sin (atan (/ (/ eh t) ew)))))
   (fabs (cbrt (pow (* ew (sin t)) 3.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -9.4e-20) || !(eh <= 7.4e-178)) {
		tmp = fabs((eh * sin(atan(((eh / t) / ew)))));
	} else {
		tmp = fabs(cbrt(pow((ew * sin(t)), 3.0)));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -9.4e-20) || !(eh <= 7.4e-178)) {
		tmp = Math.abs((eh * Math.sin(Math.atan(((eh / t) / ew)))));
	} else {
		tmp = Math.abs(Math.cbrt(Math.pow((ew * Math.sin(t)), 3.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -9.4e-20) || !(eh <= 7.4e-178))
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(eh / t) / ew)))));
	else
		tmp = abs(cbrt((Float64(ew * sin(t)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -9.4e-20], N[Not[LessEqual[eh, 7.4e-178]], $MachinePrecision]], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Power[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.4000000000000003e-20 or 7.40000000000000008e-178 < eh

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 59.6%

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

      2. pow2 59.6%

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

      3. associate-*l* 59.6%

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

      4. cos-atan 84.0%

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

      5. un-div-inv 84.0%

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

      6. hypot-1-def 80.0%

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in t around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. associate-/r* 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in t around 0 50.1%

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

   if -9.4000000000000003e-20 < eh < 7.40000000000000008e-178

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 62.7%

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

      2. pow2 62.7%

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

      3. associate-*l* 62.7%

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

      4. cos-atan 63.9%

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

      5. un-div-inv 63.9%

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

      6. hypot-1-def 63.9%

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

   4. Applied egg-rr 63.9%

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

   5. Taylor expanded in ew around inf 62.0%

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

      1. add-cbrt-cube 33.4%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot \left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \cdot \left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}}\right| \]

      2. pow3 33.4%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left({\left(\sqrt{ew \cdot \sin t}\right)}^{2} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{3}}}\right| \]

      3. +-commutative 33.4%

         \[\leadsto \left|\sqrt[3]{{\color{blue}{\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + {\left(\sqrt{ew \cdot \sin t}\right)}^{2}\right)}}^{3}}\right| \]

      4. fma-define 33.4%

         \[\leadsto \left|\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(eh \cdot \cos t, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), {\left(\sqrt{ew \cdot \sin t}\right)}^{2}\right)\right)}}^{3}}\right| \]

      5. unpow2 33.4%

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

      6. add-sqr-sqrt 53.4%

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

   7. Applied egg-rr 53.4%

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

   8. Taylor expanded in eh around 0 29.2%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3} \cdot {\sin t}^{3}}}\right| \]
   9. Step-by-step derivation

      1. cube-prod 42.9%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \sin t\right)}^{3}}}\right| \]

   10. Simplified 42.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9.4 \cdot 10^{-20} \lor \neg \left(eh \leq 7.4 \cdot 10^{-178}\right):\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{{\left(ew \cdot \sin t\right)}^{3}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 60.7%

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

   2. pow2 60.7%

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

   3. associate-*l* 60.7%

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

   4. cos-atan 77.4%

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

   5. un-div-inv 77.4%

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

   6. hypot-1-def 74.7%

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

4. Applied egg-rr 74.7%

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

5. Taylor expanded in t around 0 42.2%

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

   1. *-commutative 42.2%

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

   2. associate-/r* 42.2%

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

7. Simplified 42.2%

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

8. Taylor expanded in t around 0 41.0%

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

Alternative 12: 40.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * 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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 60.7%

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

   2. pow2 60.7%

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

   3. associate-*l* 60.7%

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

   4. cos-atan 77.4%

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

   5. un-div-inv 77.4%

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

   6. hypot-1-def 74.7%

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

4. Applied egg-rr 74.7%

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

5. Taylor expanded in t around 0 42.2%

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

   1. *-commutative 42.2%

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

   2. associate-/r* 42.2%

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

7. Simplified 42.2%

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

8. Taylor expanded in t around 0 40.9%

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

Reproduce

herbie shell --seed 2024141 
(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))))))))