Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.1s

Alternatives: 10

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. associate-/l/ 99.8%

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

   5. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

Alternative 2: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 98.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+ (* 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(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(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[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. add-cube-cbrt 99.2%

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

   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%

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

   7. associate-/l/ 99.2%

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

   8. *-commutative 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in eh around 0 98.0%

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

6. Final simplification 98.0%

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

Alternative 4: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;eh \leq -2.6 \cdot 10^{+120} \lor \neg \left(eh \leq 6.3 \cdot 10^{+86}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (/ eh ew) (tan t))))))
   (if (or (<= eh -2.6e+120) (not (<= eh 6.3e+86)))
     (fabs (* (* eh (cos t)) t_1))
     (fabs (+ (* ew (sin t)) (* eh t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((eh / ew) / tan(t))));
	double tmp;
	if ((eh <= -2.6e+120) || !(eh <= 6.3e+86)) {
		tmp = fabs(((eh * cos(t)) * t_1));
	} else {
		tmp = fabs(((ew * sin(t)) + (eh * 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 = sin(atan(((eh / ew) / tan(t))))
    if ((eh <= (-2.6d+120)) .or. (.not. (eh <= 6.3d+86))) then
        tmp = abs(((eh * cos(t)) * t_1))
    else
        tmp = abs(((ew * sin(t)) + (eh * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan(((eh / ew) / Math.tan(t))));
	double tmp;
	if ((eh <= -2.6e+120) || !(eh <= 6.3e+86)) {
		tmp = Math.abs(((eh * Math.cos(t)) * t_1));
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((eh / ew) / math.tan(t))))
	tmp = 0
	if (eh <= -2.6e+120) or not (eh <= 6.3e+86):
		tmp = math.fabs(((eh * math.cos(t)) * t_1))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(eh / ew) / tan(t))))
	tmp = 0.0
	if ((eh <= -2.6e+120) || !(eh <= 6.3e+86))
		tmp = abs(Float64(Float64(eh * cos(t)) * t_1));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((eh / ew) / tan(t))));
	tmp = 0.0;
	if ((eh <= -2.6e+120) || ~((eh <= 6.3e+86)))
		tmp = abs(((eh * cos(t)) * t_1));
	else
		tmp = abs(((ew * sin(t)) + (eh * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[eh, -2.6e+120], N[Not[LessEqual[eh, 6.3e+86]], $MachinePrecision]], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.5999999999999999e120 or 6.30000000000000023e86 < 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 98.0%

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

      2. pow3 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in eh around inf 99.8%

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

   6. Taylor expanded in eh around inf 93.8%

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

      1. associate-*r* 93.8%

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

      2. associate-/r* 93.8%

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

   8. Simplified 93.8%

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

   if -2.5999999999999999e120 < eh < 6.30000000000000023e86

   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 98.9%

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

      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%

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

      7. associate-/l/ 98.9%

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

      8. *-commutative 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in eh around 0 97.6%

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

   6. Taylor expanded in t around 0 84.9%

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

      1. associate-/r* 84.9%

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

   8. Simplified 84.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.6 \cdot 10^{+120} \lor \neg \left(eh \leq 6.3 \cdot 10^{+86}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.000235 \lor \neg \left(t \leq 7.1 \cdot 10^{-10}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.000235) (not (<= t 7.1e-10)))
   (fabs (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))
   (fabs (+ (* ew t) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.000235) || !(t <= 7.1e-10)) {
		tmp = fabs(((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))));
	} else {
		tmp = fabs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.000235d0)) .or. (.not. (t <= 7.1d-10))) then
        tmp = abs(((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))))
    else
        tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.000235) || !(t <= 7.1e-10)) {
		tmp = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))))));
	} else {
		tmp = Math.abs(((ew * t) + (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.000235) or not (t <= 7.1e-10):
		tmp = math.fabs(((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))))
	else:
		tmp = math.fabs(((ew * t) + (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.000235) || !(t <= 7.1e-10))
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))));
	else
		tmp = abs(Float64(Float64(ew * t) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.000235) || ~((t <= 7.1e-10)))
		tmp = abs(((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))));
	else
		tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.000235], N[Not[LessEqual[t, 7.1e-10]], $MachinePrecision]], N[Abs[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], 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 < -2.34999999999999993e-4 or 7.1000000000000003e-10 < 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-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in eh around inf 95.0%

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

   6. Taylor expanded in eh around inf 59.3%

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

      1. associate-*r* 59.3%

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

      2. associate-/r* 59.3%

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

   8. Simplified 59.3%

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

   if -2.34999999999999993e-4 < t < 7.1000000000000003e-10

   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-cube-cbrt 99.4%

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

      2. pow3 99.4%

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

      3. associate-*l* 99.4%

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

      4. cos-atan 99.4%

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

      5. un-div-inv 99.4%

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

      6. hypot-1-def 99.4%

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

      7. associate-/l/ 99.4%

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

      8. *-commutative 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in eh around 0 98.5%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

      2. expm1-undefine 67.7%

         \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin t\right)} - 1\right)}\right| \]

   7. Applied egg-rr 67.7%

      \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin t\right)} - 1\right)}\right| \]
   8. Step-by-step derivation

      1. expm1-define 98.5%

         \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   9. Simplified 98.5%

      \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   10. Taylor expanded in t around 0 98.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.000235 \lor \neg \left(t \leq 7.1 \cdot 10^{-10}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{+170}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.45e+170)
   (fabs (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))
   (fabs (+ (* ew (sin t)) (* eh (* (cos t) (sin (atan (/ (/ eh ew) t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.45e+170) {
		tmp = fabs(((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))));
	} else {
		tmp = fabs(((ew * sin(t)) + (eh * (cos(t) * sin(atan(((eh / ew) / 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 (eh <= (-1.45d+170)) then
        tmp = abs(((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))))
    else
        tmp = abs(((ew * sin(t)) + (eh * (cos(t) * sin(atan(((eh / ew) / t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.45e+170) {
		tmp = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))))));
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * (Math.cos(t) * Math.sin(Math.atan(((eh / ew) / t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.45e+170:
		tmp = math.fabs(((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * (math.cos(t) * math.sin(math.atan(((eh / ew) / t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.45e+170)
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh / ew) / t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.45e+170)
		tmp = abs(((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))));
	else
		tmp = abs(((ew * sin(t)) + (eh * (cos(t) * sin(atan(((eh / ew) / t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.45e+170], N[Abs[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], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.45e170

   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 98.0%

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

      2. pow3 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in eh around inf 99.6%

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

   6. Taylor expanded in eh around inf 96.3%

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

      1. associate-*r* 96.3%

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

      2. associate-/r* 96.3%

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

   8. Simplified 96.3%

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

   if -1.45e170 < 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 99.1%

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

      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%

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

      7. associate-/l/ 99.1%

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

      8. *-commutative 99.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in eh around 0 98.1%

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

   6. Taylor expanded in t around 0 91.0%

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

      1. associate-/r* 91.0%

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

   8. Simplified 91.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{+170}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \lor \neg \left(t \leq 0.0062\right):\\ \;\;\;\;\left|ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -2.1) (not (<= t 0.0062)))
   (fabs (* ew (* (sin t) (cos (atan (/ eh (* ew t)))))))
   (fabs (+ (* ew t) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2.1) || !(t <= 0.0062)) {
		tmp = fabs((ew * (sin(t) * cos(atan((eh / (ew * t)))))));
	} else {
		tmp = fabs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.1d0)) .or. (.not. (t <= 0.0062d0))) then
        tmp = abs((ew * (sin(t) * cos(atan((eh / (ew * t)))))))
    else
        tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2.1) || !(t <= 0.0062)) {
		tmp = Math.abs((ew * (Math.sin(t) * Math.cos(Math.atan((eh / (ew * t)))))));
	} else {
		tmp = Math.abs(((ew * t) + (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -2.1) or not (t <= 0.0062):
		tmp = math.fabs((ew * (math.sin(t) * math.cos(math.atan((eh / (ew * t)))))))
	else:
		tmp = math.fabs(((ew * t) + (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -2.1) || !(t <= 0.0062))
		tmp = abs(Float64(ew * Float64(sin(t) * cos(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = abs(Float64(Float64(ew * t) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -2.1) || ~((t <= 0.0062)))
		tmp = abs((ew * (sin(t) * cos(atan((eh / (ew * t)))))));
	else
		tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -2.1], N[Not[LessEqual[t, 0.0062]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $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 < -2.10000000000000009 or 0.00619999999999999978 < 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-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in eh around inf 94.9%

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

   6. Taylor expanded in eh around 0 44.8%

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

      1. *-commutative 44.8%

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

      2. associate-/r* 44.8%

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

   8. Simplified 44.8%

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

   9. Taylor expanded in t around 0 45.0%

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

   if -2.10000000000000009 < t < 0.00619999999999999978

   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-cube-cbrt 99.4%

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

      2. pow3 99.4%

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

      3. associate-*l* 99.4%

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

      4. cos-atan 99.4%

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

      5. un-div-inv 99.4%

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

      6. hypot-1-def 99.4%

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

      7. associate-/l/ 99.4%

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

      8. *-commutative 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in eh around 0 98.3%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

      2. expm1-undefine 68.2%

         \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin t\right)} - 1\right)}\right| \]

   7. Applied egg-rr 68.2%

      \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin t\right)} - 1\right)}\right| \]
   8. Step-by-step derivation

      1. expm1-define 98.3%

         \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   9. Simplified 98.3%

      \[\leadsto \left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   10. Taylor expanded in t around 0 97.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \lor \neg \left(t \leq 0.0062\right):\\ \;\;\;\;\left|ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ \mathbf{if}\;t \leq -690000 \lor \neg \left(t \leq 0.0054\right):\\ \;\;\;\;\sin t \cdot \frac{ew}{\mathsf{hypot}\left(1, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \cos \tan^{-1} t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (if (or (<= t -690000.0) (not (<= t 0.0054)))
     (* (sin t) (/ ew (hypot 1.0 t_1)))
     (fabs (* ew (* t (cos (atan t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	double tmp;
	if ((t <= -690000.0) || !(t <= 0.0054)) {
		tmp = sin(t) * (ew / hypot(1.0, t_1));
	} else {
		tmp = fabs((ew * (t * cos(atan(t_1)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	double tmp;
	if ((t <= -690000.0) || !(t <= 0.0054)) {
		tmp = Math.sin(t) * (ew / Math.hypot(1.0, t_1));
	} else {
		tmp = Math.abs((ew * (t * Math.cos(Math.atan(t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	tmp = 0
	if (t <= -690000.0) or not (t <= 0.0054):
		tmp = math.sin(t) * (ew / math.hypot(1.0, t_1))
	else:
		tmp = math.fabs((ew * (t * math.cos(math.atan(t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	tmp = 0.0
	if ((t <= -690000.0) || !(t <= 0.0054))
		tmp = Float64(sin(t) * Float64(ew / hypot(1.0, t_1)));
	else
		tmp = abs(Float64(ew * Float64(t * cos(atan(t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	tmp = 0.0;
	if ((t <= -690000.0) || ~((t <= 0.0054)))
		tmp = sin(t) * (ew / hypot(1.0, t_1));
	else
		tmp = abs((ew * (t * cos(atan(t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -690000.0], N[Not[LessEqual[t, 0.0054]], $MachinePrecision]], N[(N[Sin[t], $MachinePrecision] * N[(ew / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(ew * N[(t * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.9e5 or 0.0054000000000000003 < 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-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in eh around inf 94.9%

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

   6. Taylor expanded in eh around 0 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-/r* 44.7%

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

   8. Simplified 44.7%

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

      1. fabs-mul 44.7%

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

      2. add-sqr-sqrt 20.5%

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

      3. fabs-sqr 20.5%

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

      4. add-sqr-sqrt 21.4%

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

      5. add-sqr-sqrt 11.1%

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

      6. fabs-sqr 11.1%

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

      7. add-sqr-sqrt 21.8%

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

      8. associate-*r* 21.8%

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

      9. cos-atan 21.7%

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

      10. hypot-1-def 21.7%

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

      11. div-inv 21.7%

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

      12. *-commutative 21.7%

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

      13. associate-*r/ 21.7%

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

      14. *-commutative 21.7%

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

   10. Applied egg-rr 21.7%

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

   if -6.9e5 < t < 0.0054000000000000003

   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-cube-cbrt 98.5%

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

      2. pow3 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in eh around inf 96.5%

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

   6. Taylor expanded in eh around 0 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/r* 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in t around 0 35.2%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -690000 \lor \neg \left(t \leq 0.0054\right):\\ \;\;\;\;\sin t \cdot \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $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. Step-by-step derivation

   1. add-cube-cbrt 98.5%

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

   2. pow3 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in eh around inf 95.7%

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

6. Taylor expanded in eh around 0 39.9%

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

   1. *-commutative 39.9%

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

   2. associate-/r* 39.9%

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

8. Simplified 39.9%

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

9. Taylor expanded in t around 0 40.0%

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

Alternative 10: 21.6% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (* (sin t) (/ ew (hypot 1.0 (/ eh (* ew (tan t)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(N[Sin[t], $MachinePrecision] * N[(ew / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $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 98.5%

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

   2. pow3 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in eh around inf 95.7%

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

6. Taylor expanded in eh around 0 39.9%

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

   1. *-commutative 39.9%

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

   2. associate-/r* 39.9%

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

8. Simplified 39.9%

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

   1. fabs-mul 39.9%

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

   2. add-sqr-sqrt 19.5%

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

   3. fabs-sqr 19.5%

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

   4. add-sqr-sqrt 20.6%

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

   5. add-sqr-sqrt 8.6%

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

   6. fabs-sqr 8.6%

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

   7. add-sqr-sqrt 18.8%

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

   8. associate-*r* 18.8%

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

   9. cos-atan 18.7%

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

   10. hypot-1-def 18.7%

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

   11. div-inv 18.7%

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

   12. *-commutative 18.7%

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

   13. associate-*r/ 18.7%

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

   14. *-commutative 18.7%

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

10. Applied egg-rr 18.7%

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

11. Final simplification 18.7%

    \[\leadsto \sin t \cdot \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(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))))))))