Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.2s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. Add Preprocessing

Alternative 2: 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-cube-cbrt 99.0%

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

   2. pow3 99.1%

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

   3. associate-*l* 99.1%

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

   4. cos-atan 99.1%

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

   5. un-div-inv 99.1%

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

   6. hypot-1-def 99.1%

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

4. Applied egg-rr 99.1%

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

   1. rem-cube-cbrt 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.7%

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

   4. 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 3: 99.0% 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 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 98.4% 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 98.6%

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

Alternative 5: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;eh \leq -1.25 \cdot 10^{-74} \lor \neg \left(eh \leq 5.6 \cdot 10^{-158}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin t\_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\sin t \cdot \cos t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew (tan t))))))
   (if (or (<= eh -1.25e-74) (not (<= eh 5.6e-158)))
     (fabs (* eh (* (cos t) (sin t_1))))
     (fabs (* ew (* (sin t) (cos t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	double tmp;
	if ((eh <= -1.25e-74) || !(eh <= 5.6e-158)) {
		tmp = fabs((eh * (cos(t) * sin(t_1))));
	} else {
		tmp = fabs((ew * (sin(t) * cos(t_1))));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = atan((eh / (ew * tan(t))))
    if ((eh <= (-1.25d-74)) .or. (.not. (eh <= 5.6d-158))) then
        tmp = abs((eh * (cos(t) * sin(t_1))))
    else
        tmp = abs((ew * (sin(t) * cos(t_1))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh / (ew * Math.tan(t))));
	double tmp;
	if ((eh <= -1.25e-74) || !(eh <= 5.6e-158)) {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(t_1))));
	} else {
		tmp = Math.abs((ew * (Math.sin(t) * Math.cos(t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * math.tan(t))))
	tmp = 0
	if (eh <= -1.25e-74) or not (eh <= 5.6e-158):
		tmp = math.fabs((eh * (math.cos(t) * math.sin(t_1))))
	else:
		tmp = math.fabs((ew * (math.sin(t) * math.cos(t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	tmp = 0.0
	if ((eh <= -1.25e-74) || !(eh <= 5.6e-158))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(t_1))));
	else
		tmp = abs(Float64(ew * Float64(sin(t) * cos(t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh / (ew * tan(t))));
	tmp = 0.0;
	if ((eh <= -1.25e-74) || ~((eh <= 5.6e-158)))
		tmp = abs((eh * (cos(t) * sin(t_1))));
	else
		tmp = abs((ew * (sin(t) * cos(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[eh, -1.25e-74], N[Not[LessEqual[eh, 5.6e-158]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.25e-74 or 5.60000000000000004e-158 < 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-cube-cbrt 99.5%

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

      2. pow3 99.5%

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

      3. associate-*l* 99.5%

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

      4. cos-atan 99.5%

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

      5. un-div-inv 99.5%

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

      6. hypot-1-def 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ew around 0 83.8%

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

   if -1.25e-74 < eh < 5.60000000000000004e-158

   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-cbrt-cube 85.2%

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

      2. pow3 85.2%

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in ew around inf 83.0%

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

      1. *-commutative 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.25 \cdot 10^{-74} \lor \neg \left(eh \leq 5.6 \cdot 10^{-158}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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-cube-cbrt 99.0%

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

   2. pow3 99.1%

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

   3. associate-*l* 99.1%

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

   4. cos-atan 99.1%

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

   5. un-div-inv 99.1%

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

   6. hypot-1-def 99.1%

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

4. Applied egg-rr 99.1%

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

5. Taylor expanded in ew around 0 63.5%

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

Alternative 7: 48.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \mathbf{if}\;t \leq -0.00021 \lor \neg \left(t \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t}{t\_1} + eh \cdot \sin \tan^{-1} t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))))
   (if (or (<= t -0.00021) (not (<= t 7.5e+76)))
     (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))
     (fabs (+ (/ (* ew (sin t)) t_1) (* eh (sin (atan t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double tmp;
	if ((t <= -0.00021) || !(t <= 7.5e+76)) {
		tmp = (eh * cos(t)) * sin(atan(((eh / ew) / tan(t))));
	} else {
		tmp = fabs((((ew * sin(t)) / t_1) + (eh * sin(atan(t_1)))));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = eh / (ew * t)
    if ((t <= (-0.00021d0)) .or. (.not. (t <= 7.5d+76))) then
        tmp = (eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))
    else
        tmp = abs((((ew * sin(t)) / t_1) + (eh * sin(atan(t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double tmp;
	if ((t <= -0.00021) || !(t <= 7.5e+76)) {
		tmp = (eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))));
	} else {
		tmp = Math.abs((((ew * Math.sin(t)) / t_1) + (eh * Math.sin(Math.atan(t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	tmp = 0
	if (t <= -0.00021) or not (t <= 7.5e+76):
		tmp = (eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))
	else:
		tmp = math.fabs((((ew * math.sin(t)) / t_1) + (eh * math.sin(math.atan(t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	tmp = 0.0
	if ((t <= -0.00021) || !(t <= 7.5e+76))
		tmp = Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t)))));
	else
		tmp = abs(Float64(Float64(Float64(ew * sin(t)) / t_1) + Float64(eh * sin(atan(t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	tmp = 0.0;
	if ((t <= -0.00021) || ~((t <= 7.5e+76)))
		tmp = (eh * cos(t)) * sin(atan(((eh / ew) / tan(t))));
	else
		tmp = abs((((ew * sin(t)) / t_1) + (eh * sin(atan(t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.00021], N[Not[LessEqual[t, 7.5e+76]], $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], N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000001e-4 or 7.4999999999999995e76 < t

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.9%

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

      3. associate-*l* 98.9%

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

      4. cos-atan 98.9%

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

      5. un-div-inv 98.9%

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

      6. hypot-1-def 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. rem-cube-cbrt 99.7%

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

      2. add-sqr-sqrt 46.0%

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

      3. fabs-sqr 46.0%

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

      4. add-sqr-sqrt 47.0%

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

   6. Applied egg-rr 47.0%

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

   7. Taylor expanded in eh around inf 25.9%

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

      1. associate-*r* 25.9%

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

      2. associate-/r* 25.9%

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

   9. Simplified 25.9%

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

   if -2.1000000000000001e-4 < t < 7.4999999999999995e76

   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. cos-atan 99.9%

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

      2. un-div-inv 99.9%

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

      3. hypot-1-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 68.7%

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

   6. Taylor expanded in t around 0 64.0%

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

   7. Taylor expanded in t around 0 64.0%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00021 \lor \neg \left(t \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t}{\frac{eh}{ew \cdot t}} + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.01 \lor \neg \left(t \leq 2.1 \cdot 10^{+78}\right):\\ \;\;\;\;\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.01) (not (<= t 2.1e+78)))
   (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))
   (fabs eh)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.01) || !(t <= 2.1e+78)) {
		tmp = (eh * cos(t)) * sin(atan(((eh / ew) / tan(t))));
	} else {
		tmp = fabs(eh);
	}
	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 <= (-0.01d0)) .or. (.not. (t <= 2.1d+78))) then
        tmp = (eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.01) || !(t <= 2.1e+78)) {
		tmp = (eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.01) or not (t <= 2.1e+78):
		tmp = (eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.01) || !(t <= 2.1e+78))
		tmp = Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t)))));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.01) || ~((t <= 2.1e+78)))
		tmp = (eh * cos(t)) * sin(atan(((eh / ew) / tan(t))));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.01], N[Not[LessEqual[t, 2.1e+78]], $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], N[Abs[eh], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0100000000000000002 or 2.1000000000000001e78 < t

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.9%

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

      3. associate-*l* 98.9%

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

      4. cos-atan 98.9%

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

      5. un-div-inv 98.9%

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

      6. hypot-1-def 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. rem-cube-cbrt 99.7%

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

      2. add-sqr-sqrt 46.0%

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

      3. fabs-sqr 46.0%

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

      4. add-sqr-sqrt 47.0%

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

   6. Applied egg-rr 47.0%

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

   7. Taylor expanded in eh around inf 25.9%

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

      1. associate-*r* 25.9%

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

      2. associate-/r* 25.9%

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

   9. Simplified 25.9%

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

   if -0.0100000000000000002 < t < 2.1000000000000001e78

   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-cube-cbrt 99.2%

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

      2. pow3 99.2%

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

      3. associate-*l* 99.2%

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

      4. cos-atan 99.2%

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

      5. un-div-inv 99.2%

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

      6. hypot-1-def 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in t around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. sin-atan 21.6%

         \[\leadsto \left|eh \cdot \color{blue}{\frac{\frac{eh}{\tan t \cdot ew}}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}}\right| \]

      3. div-inv 21.5%

         \[\leadsto \left|eh \cdot \frac{\color{blue}{eh \cdot \frac{1}{\tan t \cdot ew}}}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}\right| \]

      4. *-commutative 21.5%

         \[\leadsto \left|eh \cdot \frac{eh \cdot \frac{1}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}\right| \]

      5. hypot-1-def 32.3%

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

      6. *-un-lft-identity 32.3%

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

      7. times-frac 32.4%

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

      8. *-commutative 32.4%

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in eh around inf 63.7%

      \[\leadsto \left|\color{blue}{eh}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.01 \lor \neg \left(t \leq 2.1 \cdot 10^{+78}\right):\\ \;\;\;\;\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|eh\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(eh)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $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-cube-cbrt 99.0%

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

   2. pow3 99.1%

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

   3. associate-*l* 99.1%

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

   4. cos-atan 99.1%

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

   5. un-div-inv 99.1%

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

   6. hypot-1-def 99.1%

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

4. Applied egg-rr 99.1%

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

5. Taylor expanded in t around 0 44.0%

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

   1. *-commutative 44.0%

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

   2. sin-atan 16.5%

      \[\leadsto \left|eh \cdot \color{blue}{\frac{\frac{eh}{\tan t \cdot ew}}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}}\right| \]

   3. div-inv 16.4%

      \[\leadsto \left|eh \cdot \frac{\color{blue}{eh \cdot \frac{1}{\tan t \cdot ew}}}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}\right| \]

   4. *-commutative 16.4%

      \[\leadsto \left|eh \cdot \frac{eh \cdot \frac{1}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}\right| \]

   5. hypot-1-def 24.4%

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

   6. *-un-lft-identity 24.4%

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

   7. times-frac 24.5%

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

   8. *-commutative 24.5%

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

7. Applied egg-rr 24.5%

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

8. Taylor expanded in eh around inf 44.4%

   \[\leadsto \left|\color{blue}{eh}\right| \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))))))