Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.6s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(((eh * (cos(t) * sin(t_1))) + (ew * (cos(t_1) * sin(t)))))
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(((eh * (Math.cos(t) * Math.sin(t_1))) + (ew * (Math.cos(t_1) * Math.sin(t)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

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. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

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 3: 99.7% accurate, 1.1× speedup

Math

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

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(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t))))))) + Float64(ew / Float64(hypot(1.0, Float64(Float64(eh / 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[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ew / N[(N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $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. Step-by-step derivation

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r* 99.8%

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

   3. cos-atan 99.8%

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

   4. hypot-1-def 99.8%

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

   5. div-inv 99.8%

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

   6. clear-num 99.7%

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

   7. un-div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r* 99.8%

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

   3. cos-atan 99.8%

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

   4. hypot-1-def 99.8%

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

   5. div-inv 99.8%

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

   6. clear-num 99.7%

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

   7. un-div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in ew around inf 98.4%

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

Alternative 5: 56.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;eh \leq -7 \cdot 10^{+128}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right)\\ \mathbf{elif}\;eh \leq -0.27:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} + -0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew}}{t}\right)\right|\\ \mathbf{elif}\;eh \leq 94:\\ \;\;\;\;\left|ew \cdot \left(\cos t\_1 \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew (tan t))))))
   (if (<= eh -7e+128)
     (* eh (* (cos t) (sin (atan (/ (/ eh (tan t)) ew)))))
     (if (<= eh -0.27)
       (fabs
        (*
         eh
         (sin
          (atan
           (/
            (+ (/ eh ew) (* -0.3333333333333333 (/ (* eh (pow t 2.0)) ew)))
            t)))))
       (if (<= eh 94.0)
         (fabs (* ew (* (cos t_1) (sin t))))
         (fabs (* eh (sin t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	double tmp;
	if (eh <= -7e+128) {
		tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))));
	} else if (eh <= -0.27) {
		tmp = fabs((eh * sin(atan((((eh / ew) + (-0.3333333333333333 * ((eh * pow(t, 2.0)) / ew))) / t)))));
	} else if (eh <= 94.0) {
		tmp = fabs((ew * (cos(t_1) * sin(t))));
	} else {
		tmp = fabs((eh * sin(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 <= (-7d+128)) then
        tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))))
    else if (eh <= (-0.27d0)) then
        tmp = abs((eh * sin(atan((((eh / ew) + ((-0.3333333333333333d0) * ((eh * (t ** 2.0d0)) / ew))) / t)))))
    else if (eh <= 94.0d0) then
        tmp = abs((ew * (cos(t_1) * sin(t))))
    else
        tmp = abs((eh * sin(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 <= -7e+128) {
		tmp = eh * (Math.cos(t) * Math.sin(Math.atan(((eh / Math.tan(t)) / ew))));
	} else if (eh <= -0.27) {
		tmp = Math.abs((eh * Math.sin(Math.atan((((eh / ew) + (-0.3333333333333333 * ((eh * Math.pow(t, 2.0)) / ew))) / t)))));
	} else if (eh <= 94.0) {
		tmp = Math.abs((ew * (Math.cos(t_1) * Math.sin(t))));
	} else {
		tmp = Math.abs((eh * Math.sin(t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * math.tan(t))))
	tmp = 0
	if eh <= -7e+128:
		tmp = eh * (math.cos(t) * math.sin(math.atan(((eh / math.tan(t)) / ew))))
	elif eh <= -0.27:
		tmp = math.fabs((eh * math.sin(math.atan((((eh / ew) + (-0.3333333333333333 * ((eh * math.pow(t, 2.0)) / ew))) / t)))))
	elif eh <= 94.0:
		tmp = math.fabs((ew * (math.cos(t_1) * math.sin(t))))
	else:
		tmp = math.fabs((eh * math.sin(t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	tmp = 0.0
	if (eh <= -7e+128)
		tmp = Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh / tan(t)) / ew)))));
	elseif (eh <= -0.27)
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(Float64(eh / ew) + Float64(-0.3333333333333333 * Float64(Float64(eh * (t ^ 2.0)) / ew))) / t)))));
	elseif (eh <= 94.0)
		tmp = abs(Float64(ew * Float64(cos(t_1) * sin(t))));
	else
		tmp = abs(Float64(eh * sin(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 <= -7e+128)
		tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))));
	elseif (eh <= -0.27)
		tmp = abs((eh * sin(atan((((eh / ew) + (-0.3333333333333333 * ((eh * (t ^ 2.0)) / ew))) / t)))));
	elseif (eh <= 94.0)
		tmp = abs((ew * (cos(t_1) * sin(t))));
	else
		tmp = abs((eh * sin(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[LessEqual[eh, -7e+128], N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eh, -0.27], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(N[(eh / ew), $MachinePrecision] + N[(-0.3333333333333333 * N[(N[(eh * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 94.0], N[Abs[N[(ew * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if eh < -6.99999999999999937e128

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in ew around 0 57.2%

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

      1. rem-cube-cbrt 58.5%

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

      2. *-commutative 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-/r* 58.5%

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

   9. Applied egg-rr 58.5%

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

   if -6.99999999999999937e128 < eh < -0.27000000000000002

   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. Step-by-step derivation

      1. fma-define 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 64.3%

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

   6. Taylor expanded in t around 0 64.4%

      \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

   if -0.27000000000000002 < eh < 94

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around inf 99.8%

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

   6. Taylor expanded in ew around inf 67.0%

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

      1. *-commutative 67.0%

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

   8. Simplified 67.0%

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

   if 94 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 59.1%

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}\right| \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{+128}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right)\\ \mathbf{elif}\;eh \leq -0.27:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} + -0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew}}{t}\right)\right|\\ \mathbf{elif}\;eh \leq 94:\\ \;\;\;\;\left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;eh \leq -9.5 \cdot 10^{+37} \lor \neg \left(eh \leq 8.5 \cdot 10^{+34}\right):\\ \;\;\;\;\left|t\_1 \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ eh (* ew (tan t)))))))
   (if (or (<= eh -9.5e+37) (not (<= eh 8.5e+34)))
     (fabs (* t_1 (* eh (cos t))))
     (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 <= -9.5e+37) || !(eh <= 8.5e+34)) {
		tmp = fabs((t_1 * (eh * cos(t))));
	} 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 <= (-9.5d+37)) .or. (.not. (eh <= 8.5d+34))) then
        tmp = abs((t_1 * (eh * cos(t))))
    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 <= -9.5e+37) || !(eh <= 8.5e+34)) {
		tmp = Math.abs((t_1 * (eh * Math.cos(t))));
	} 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 <= -9.5e+37) or not (eh <= 8.5e+34):
		tmp = math.fabs((t_1 * (eh * math.cos(t))))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(eh / Float64(ew * tan(t)))))
	tmp = 0.0
	if ((eh <= -9.5e+37) || !(eh <= 8.5e+34))
		tmp = abs(Float64(t_1 * Float64(eh * cos(t))));
	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 <= -9.5e+37) || ~((eh <= 8.5e+34)))
		tmp = abs((t_1 * (eh * cos(t))));
	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[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[eh, -9.5e+37], N[Not[LessEqual[eh, 8.5e+34]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $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 < -9.4999999999999995e37 or 8.5000000000000003e34 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 91.2%

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

      1. associate-*r* 91.2%

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

      2. *-commutative 91.2%

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

   7. Simplified 91.2%

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

   if -9.4999999999999995e37 < eh < 8.5000000000000003e34

   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. Step-by-step derivation

      1. fma-define 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in ew around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.9%

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

      3. cos-atan 99.9%

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

      4. hypot-1-def 99.9%

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

      5. div-inv 99.9%

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

      6. clear-num 99.7%

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

      7. un-div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ew around inf 98.9%

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

   9. Taylor expanded in t around 0 89.6%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9.5 \cdot 10^{+37} \lor \neg \left(eh \leq 8.5 \cdot 10^{+34}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;eh \leq -7.2 \cdot 10^{-117} \lor \neg \left(eh \leq 16\right):\\ \;\;\;\;\left|\sin t\_1 \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t\_1 \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew (tan t))))))
   (if (or (<= eh -7.2e-117) (not (<= eh 16.0)))
     (fabs (* (sin t_1) (* eh (cos t))))
     (fabs (* ew (* (cos t_1) (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	double tmp;
	if ((eh <= -7.2e-117) || !(eh <= 16.0)) {
		tmp = fabs((sin(t_1) * (eh * cos(t))));
	} else {
		tmp = fabs((ew * (cos(t_1) * sin(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) :: t_1
    real(8) :: tmp
    t_1 = atan((eh / (ew * tan(t))))
    if ((eh <= (-7.2d-117)) .or. (.not. (eh <= 16.0d0))) then
        tmp = abs((sin(t_1) * (eh * cos(t))))
    else
        tmp = abs((ew * (cos(t_1) * sin(t))))
    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 <= -7.2e-117) || !(eh <= 16.0)) {
		tmp = Math.abs((Math.sin(t_1) * (eh * Math.cos(t))));
	} else {
		tmp = Math.abs((ew * (Math.cos(t_1) * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * math.tan(t))))
	tmp = 0
	if (eh <= -7.2e-117) or not (eh <= 16.0):
		tmp = math.fabs((math.sin(t_1) * (eh * math.cos(t))))
	else:
		tmp = math.fabs((ew * (math.cos(t_1) * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	tmp = 0.0
	if ((eh <= -7.2e-117) || !(eh <= 16.0))
		tmp = abs(Float64(sin(t_1) * Float64(eh * cos(t))));
	else
		tmp = abs(Float64(ew * Float64(cos(t_1) * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh / (ew * tan(t))));
	tmp = 0.0;
	if ((eh <= -7.2e-117) || ~((eh <= 16.0)))
		tmp = abs((sin(t_1) * (eh * cos(t))));
	else
		tmp = abs((ew * (cos(t_1) * sin(t))));
	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, -7.2e-117], N[Not[LessEqual[eh, 16.0]], $MachinePrecision]], N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.2000000000000001e-117 or 16 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 82.6%

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

      1. associate-*r* 82.6%

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

      2. *-commutative 82.6%

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

   7. Simplified 82.6%

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

   if -7.2000000000000001e-117 < eh < 16

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around inf 99.8%

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

   6. Taylor expanded in ew around inf 73.7%

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

      1. *-commutative 73.7%

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

   8. Simplified 73.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.2 \cdot 10^{-117} \lor \neg \left(eh \leq 16\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+128}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right)\\ \mathbf{elif}\;eh \leq -2.9 \cdot 10^{-123}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} + -0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew}}{t}\right)\right|\\ \mathbf{elif}\;eh \leq 66:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -9e+128)
   (* eh (* (cos t) (sin (atan (/ (/ eh (tan t)) ew)))))
   (if (<= eh -2.9e-123)
     (fabs
      (*
       eh
       (sin
        (atan
         (/
          (+ (/ eh ew) (* -0.3333333333333333 (/ (* eh (pow t 2.0)) ew)))
          t)))))
     (if (<= eh 66.0)
       (sqrt (pow (* ew (sin t)) 2.0))
       (fabs (* eh (sin (atan (/ eh (* ew (tan t)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -9e+128) {
		tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))));
	} else if (eh <= -2.9e-123) {
		tmp = fabs((eh * sin(atan((((eh / ew) + (-0.3333333333333333 * ((eh * pow(t, 2.0)) / ew))) / t)))));
	} else if (eh <= 66.0) {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	} else {
		tmp = fabs((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 (eh <= (-9d+128)) then
        tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))))
    else if (eh <= (-2.9d-123)) then
        tmp = abs((eh * sin(atan((((eh / ew) + ((-0.3333333333333333d0) * ((eh * (t ** 2.0d0)) / ew))) / t)))))
    else if (eh <= 66.0d0) then
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    else
        tmp = abs((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 (eh <= -9e+128) {
		tmp = eh * (Math.cos(t) * Math.sin(Math.atan(((eh / Math.tan(t)) / ew))));
	} else if (eh <= -2.9e-123) {
		tmp = Math.abs((eh * Math.sin(Math.atan((((eh / ew) + (-0.3333333333333333 * ((eh * Math.pow(t, 2.0)) / ew))) / t)))));
	} else if (eh <= 66.0) {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -9e+128:
		tmp = eh * (math.cos(t) * math.sin(math.atan(((eh / math.tan(t)) / ew))))
	elif eh <= -2.9e-123:
		tmp = math.fabs((eh * math.sin(math.atan((((eh / ew) + (-0.3333333333333333 * ((eh * math.pow(t, 2.0)) / ew))) / t)))))
	elif eh <= 66.0:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	else:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -9e+128)
		tmp = Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh / tan(t)) / ew)))));
	elseif (eh <= -2.9e-123)
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(Float64(eh / ew) + Float64(-0.3333333333333333 * Float64(Float64(eh * (t ^ 2.0)) / ew))) / t)))));
	elseif (eh <= 66.0)
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	else
		tmp = abs(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 (eh <= -9e+128)
		tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))));
	elseif (eh <= -2.9e-123)
		tmp = abs((eh * sin(atan((((eh / ew) + (-0.3333333333333333 * ((eh * (t ^ 2.0)) / ew))) / t)))));
	elseif (eh <= 66.0)
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	else
		tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -9e+128], N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eh, -2.9e-123], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(N[(eh / ew), $MachinePrecision] + N[(-0.3333333333333333 * N[(N[(eh * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 66.0], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eh < -9.0000000000000003e128

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in ew around 0 57.2%

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

      1. rem-cube-cbrt 58.5%

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

      2. *-commutative 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-/r* 58.5%

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

   9. Applied egg-rr 58.5%

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

   if -9.0000000000000003e128 < eh < -2.90000000000000004e-123

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 51.8%

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

   6. Taylor expanded in t around 0 52.0%

      \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

   if -2.90000000000000004e-123 < eh < 66

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in ew around inf 41.9%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 40.6%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 51.4%

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

      3. pow2 51.4%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   if 66 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 59.1%

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}\right| \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+128}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right)\\ \mathbf{elif}\;eh \leq -2.9 \cdot 10^{-123}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} + -0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew}}{t}\right)\right|\\ \mathbf{elif}\;eh \leq 66:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+128}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right)\\ \mathbf{elif}\;eh \leq -5.1 \cdot 10^{-118} \lor \neg \left(eh \leq 33\right):\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -9e+128)
   (* eh (* (cos t) (sin (atan (/ (/ eh (tan t)) ew)))))
   (if (or (<= eh -5.1e-118) (not (<= eh 33.0)))
     (fabs (* eh (sin (atan (/ eh (* ew (tan t)))))))
     (sqrt (pow (* ew (sin t)) 2.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -9e+128) {
		tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))));
	} else if ((eh <= -5.1e-118) || !(eh <= 33.0)) {
		tmp = fabs((eh * sin(atan((eh / (ew * tan(t)))))));
	} else {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	}
	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 <= (-9d+128)) then
        tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))))
    else if ((eh <= (-5.1d-118)) .or. (.not. (eh <= 33.0d0))) then
        tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))))
    else
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -9e+128) {
		tmp = eh * (Math.cos(t) * Math.sin(Math.atan(((eh / Math.tan(t)) / ew))));
	} else if ((eh <= -5.1e-118) || !(eh <= 33.0)) {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	} else {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -9e+128:
		tmp = eh * (math.cos(t) * math.sin(math.atan(((eh / math.tan(t)) / ew))))
	elif (eh <= -5.1e-118) or not (eh <= 33.0):
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	else:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -9e+128)
		tmp = Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh / tan(t)) / ew)))));
	elseif ((eh <= -5.1e-118) || !(eh <= 33.0))
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	else
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -9e+128)
		tmp = eh * (cos(t) * sin(atan(((eh / tan(t)) / ew))));
	elseif ((eh <= -5.1e-118) || ~((eh <= 33.0)))
		tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))));
	else
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -9e+128], N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[eh, -5.1e-118], N[Not[LessEqual[eh, 33.0]], $MachinePrecision]], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -9.0000000000000003e128

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in ew around 0 57.2%

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

      1. rem-cube-cbrt 58.5%

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

      2. *-commutative 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-/r* 58.5%

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

   9. Applied egg-rr 58.5%

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

   if -9.0000000000000003e128 < eh < -5.09999999999999964e-118 or 33 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 55.4%

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

   if -5.09999999999999964e-118 < eh < 33

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in ew around inf 41.9%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 40.6%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 51.4%

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

      3. pow2 51.4%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 51.4%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+128}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\right)\\ \mathbf{elif}\;eh \leq -5.1 \cdot 10^{-118} \lor \neg \left(eh \leq 33\right):\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -5.2e+102)
   (sqrt (pow (* ew (sin t)) 2.0))
   (if (<= ew 5.6e-30)
     (fabs (* eh (sin (atan (/ eh (* ew (tan t)))))))
     (* ew (sqrt (pow (sin t) 2.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -5.2e+102) {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	} else if (ew <= 5.6e-30) {
		tmp = fabs((eh * sin(atan((eh / (ew * tan(t)))))));
	} else {
		tmp = ew * sqrt(pow(sin(t), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-5.2d+102)) then
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    else if (ew <= 5.6d-30) then
        tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))))
    else
        tmp = ew * sqrt((sin(t) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -5.2e+102) {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	} else if (ew <= 5.6e-30) {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	} else {
		tmp = ew * Math.sqrt(Math.pow(Math.sin(t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -5.2e+102:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	elif ew <= 5.6e-30:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	else:
		tmp = ew * math.sqrt(math.pow(math.sin(t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -5.2e+102)
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	elseif (ew <= 5.6e-30)
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	else
		tmp = Float64(ew * sqrt((sin(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -5.2e+102)
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	elseif (ew <= 5.6e-30)
		tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))));
	else
		tmp = ew * sqrt((sin(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -5.2e+102], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 5.6e-30], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew * N[Sqrt[N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -5.20000000000000013e102

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in ew around inf 35.6%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 44.1%

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

      3. pow2 44.1%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   if -5.20000000000000013e102 < ew < 5.59999999999999977e-30

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.2%

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

   if 5.59999999999999977e-30 < ew

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in ew around inf 36.5%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-unprod 54.2%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\sin t \cdot \sin t}} \]

      3. pow2 54.2%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\sin t}^{2}}} \]

   9. Applied egg-rr 54.2%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\sin t}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 48.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.75 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.75e+102)
   (sqrt (pow (* ew (sin t)) 2.0))
   (if (<= ew 1.35e-24)
     (fabs (* eh (sin (atan (/ (/ eh ew) t)))))
     (* ew (sqrt (pow (sin t) 2.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.75e+102) {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	} else if (ew <= 1.35e-24) {
		tmp = fabs((eh * sin(atan(((eh / ew) / t)))));
	} else {
		tmp = ew * sqrt(pow(sin(t), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1.75d+102)) then
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    else if (ew <= 1.35d-24) then
        tmp = abs((eh * sin(atan(((eh / ew) / t)))))
    else
        tmp = ew * sqrt((sin(t) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.75e+102) {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	} else if (ew <= 1.35e-24) {
		tmp = Math.abs((eh * Math.sin(Math.atan(((eh / ew) / t)))));
	} else {
		tmp = ew * Math.sqrt(Math.pow(Math.sin(t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.75e+102:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	elif ew <= 1.35e-24:
		tmp = math.fabs((eh * math.sin(math.atan(((eh / ew) / t)))))
	else:
		tmp = ew * math.sqrt(math.pow(math.sin(t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.75e+102)
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	elseif (ew <= 1.35e-24)
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(eh / ew) / t)))));
	else
		tmp = Float64(ew * sqrt((sin(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.75e+102)
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	elseif (ew <= 1.35e-24)
		tmp = abs((eh * sin(atan(((eh / ew) / t)))));
	else
		tmp = ew * sqrt((sin(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.75e+102], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.35e-24], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew * N[Sqrt[N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.75000000000000005e102

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in ew around inf 35.6%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 44.1%

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

      3. pow2 44.1%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   if -1.75000000000000005e102 < ew < 1.35000000000000003e-24

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.2%

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

   6. Taylor expanded in t around 0 51.6%

      \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

   7. Taylor expanded in t around 0 51.6%

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

   if 1.35000000000000003e-24 < ew

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in ew around inf 36.5%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-unprod 54.2%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\sin t \cdot \sin t}} \]

      3. pow2 54.2%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\sin t}^{2}}} \]

   9. Applied egg-rr 54.2%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\sin t}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 48.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.5 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -5.5e+101)
   (sqrt (pow (* ew (sin t)) 2.0))
   (if (<= ew 2.9e-25)
     (fabs (* eh (sin (atan (/ eh (* t ew))))))
     (* ew (sqrt (pow (sin t) 2.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -5.5e+101) {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	} else if (ew <= 2.9e-25) {
		tmp = fabs((eh * sin(atan((eh / (t * ew))))));
	} else {
		tmp = ew * sqrt(pow(sin(t), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-5.5d+101)) then
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    else if (ew <= 2.9d-25) then
        tmp = abs((eh * sin(atan((eh / (t * ew))))))
    else
        tmp = ew * sqrt((sin(t) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -5.5e+101) {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	} else if (ew <= 2.9e-25) {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (t * ew))))));
	} else {
		tmp = ew * Math.sqrt(Math.pow(Math.sin(t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -5.5e+101:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	elif ew <= 2.9e-25:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (t * ew))))))
	else:
		tmp = ew * math.sqrt(math.pow(math.sin(t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -5.5e+101)
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	elseif (ew <= 2.9e-25)
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(t * ew))))));
	else
		tmp = Float64(ew * sqrt((sin(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -5.5e+101)
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	elseif (ew <= 2.9e-25)
		tmp = abs((eh * sin(atan((eh / (t * ew))))));
	else
		tmp = ew * sqrt((sin(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -5.5e+101], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.9e-25], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew * N[Sqrt[N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -5.50000000000000018e101

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in ew around inf 35.6%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 44.1%

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

      3. pow2 44.1%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   if -5.50000000000000018e101 < ew < 2.9000000000000001e-25

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.2%

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

   6. Taylor expanded in t around 0 51.6%

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

   if 2.9000000000000001e-25 < ew

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in ew around inf 36.5%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-unprod 54.2%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\sin t \cdot \sin t}} \]

      3. pow2 54.2%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\sin t}^{2}}} \]

   9. Applied egg-rr 54.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.5 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -0.0088 \lor \neg \left(eh \leq 2.4 \cdot 10^{+68}\right):\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -0.0088) (not (<= eh 2.4e+68)))
   (* eh (sin (atan (/ (/ eh ew) t))))
   (sqrt (pow (* ew (sin t)) 2.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -0.0088) || !(eh <= 2.4e+68)) {
		tmp = eh * sin(atan(((eh / ew) / t)));
	} else {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	}
	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 <= (-0.0088d0)) .or. (.not. (eh <= 2.4d+68))) then
        tmp = eh * sin(atan(((eh / ew) / t)))
    else
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -0.0088) || !(eh <= 2.4e+68)) {
		tmp = eh * Math.sin(Math.atan(((eh / ew) / t)));
	} else {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -0.0088) or not (eh <= 2.4e+68):
		tmp = eh * math.sin(math.atan(((eh / ew) / t)))
	else:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -0.0088) || !(eh <= 2.4e+68))
		tmp = Float64(eh * sin(atan(Float64(Float64(eh / ew) / t))));
	else
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -0.0088) || ~((eh <= 2.4e+68)))
		tmp = eh * sin(atan(((eh / ew) / t)));
	else
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -0.0088], N[Not[LessEqual[eh, 2.4e+68]], $MachinePrecision]], N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -0.00880000000000000053 or 2.40000000000000008e68 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.8%

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

      1. add-cube-cbrt 53.8%

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

      2. pow3 53.8%

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

      3. add-sqr-sqrt 28.6%

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

      4. fabs-sqr 28.6%

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

      5. add-sqr-sqrt 28.9%

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

      6. associate-/r* 28.9%

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

   7. Applied egg-rr 28.9%

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

   8. Taylor expanded in t around 0 28.3%

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

      1. rem-cube-cbrt 28.8%

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

      2. *-commutative 28.8%

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

      3. associate-/r* 28.8%

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

   10. Applied egg-rr 28.8%

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

   if -0.00880000000000000053 < eh < 2.40000000000000008e68

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.7%

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

   6. Applied egg-rr 44.5%

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

   7. Taylor expanded in ew around inf 34.6%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 33.4%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 45.3%

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

      3. pow2 45.3%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 45.3%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -0.0088 \lor \neg \left(eh \leq 2.4 \cdot 10^{+68}\right):\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.1 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 9.5 \cdot 10^{-97}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.1e-85)
   (sqrt (pow (* ew (sin t)) 2.0))
   (if (<= ew 9.5e-97)
     (* eh (sin (atan (/ (/ eh ew) t))))
     (* ew (sqrt (pow (sin t) 2.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.1e-85) {
		tmp = sqrt(pow((ew * sin(t)), 2.0));
	} else if (ew <= 9.5e-97) {
		tmp = eh * sin(atan(((eh / ew) / t)));
	} else {
		tmp = ew * sqrt(pow(sin(t), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1.1d-85)) then
        tmp = sqrt(((ew * sin(t)) ** 2.0d0))
    else if (ew <= 9.5d-97) then
        tmp = eh * sin(atan(((eh / ew) / t)))
    else
        tmp = ew * sqrt((sin(t) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.1e-85) {
		tmp = Math.sqrt(Math.pow((ew * Math.sin(t)), 2.0));
	} else if (ew <= 9.5e-97) {
		tmp = eh * Math.sin(Math.atan(((eh / ew) / t)));
	} else {
		tmp = ew * Math.sqrt(Math.pow(Math.sin(t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.1e-85:
		tmp = math.sqrt(math.pow((ew * math.sin(t)), 2.0))
	elif ew <= 9.5e-97:
		tmp = eh * math.sin(math.atan(((eh / ew) / t)))
	else:
		tmp = ew * math.sqrt(math.pow(math.sin(t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.1e-85)
		tmp = sqrt((Float64(ew * sin(t)) ^ 2.0));
	elseif (ew <= 9.5e-97)
		tmp = Float64(eh * sin(atan(Float64(Float64(eh / ew) / t))));
	else
		tmp = Float64(ew * sqrt((sin(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.1e-85)
		tmp = sqrt(((ew * sin(t)) ^ 2.0));
	elseif (ew <= 9.5e-97)
		tmp = eh * sin(atan(((eh / ew) / t)));
	else
		tmp = ew * sqrt((sin(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.1e-85], N[Sqrt[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 9.5e-97], N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(ew * N[Sqrt[N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.1e-85

   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. Step-by-step derivation

      1. fma-define 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 40.9%

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

   7. Taylor expanded in ew around inf 25.6%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 24.8%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 42.4%

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

      3. pow2 42.4%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   if -1.1e-85 < ew < 9.5000000000000001e-97

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 60.3%

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

      1. add-cube-cbrt 59.3%

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

      2. pow3 59.3%

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

      3. add-sqr-sqrt 27.5%

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

      4. fabs-sqr 27.5%

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

      5. add-sqr-sqrt 28.7%

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

      6. associate-/r* 28.7%

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

   7. Applied egg-rr 28.7%

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

   8. Taylor expanded in t around 0 26.9%

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

      1. rem-cube-cbrt 27.3%

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

      2. *-commutative 27.3%

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

      3. associate-/r* 27.3%

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

   10. Applied egg-rr 27.3%

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

   if 9.5000000000000001e-97 < ew

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.5%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in ew around inf 34.9%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 34.2%

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

      2. sqrt-unprod 51.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\sin t \cdot \sin t}} \]

      3. pow2 51.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\sin t}^{2}}} \]

   9. Applied egg-rr 51.3%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.1 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 9.5 \cdot 10^{-97}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \mathbf{if}\;eh \leq -0.52:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq -5.6 \cdot 10^{-229}:\\ \;\;\;\;ew \cdot \left(-\sin t\right)\\ \mathbf{elif}\;eh \leq 2.05 \cdot 10^{+16}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin (atan (/ (/ eh ew) t))))))
   (if (<= eh -0.52)
     t_1
     (if (<= eh -5.6e-229)
       (* ew (- (sin t)))
       (if (<= eh 2.05e+16) (* ew (sin t)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(atan(((eh / ew) / t)));
	double tmp;
	if (eh <= -0.52) {
		tmp = t_1;
	} else if (eh <= -5.6e-229) {
		tmp = ew * -sin(t);
	} else if (eh <= 2.05e+16) {
		tmp = ew * sin(t);
	} else {
		tmp = 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 * sin(atan(((eh / ew) / t)))
    if (eh <= (-0.52d0)) then
        tmp = t_1
    else if (eh <= (-5.6d-229)) then
        tmp = ew * -sin(t)
    else if (eh <= 2.05d+16) then
        tmp = ew * sin(t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(Math.atan(((eh / ew) / t)));
	double tmp;
	if (eh <= -0.52) {
		tmp = t_1;
	} else if (eh <= -5.6e-229) {
		tmp = ew * -Math.sin(t);
	} else if (eh <= 2.05e+16) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(math.atan(((eh / ew) / t)))
	tmp = 0
	if eh <= -0.52:
		tmp = t_1
	elif eh <= -5.6e-229:
		tmp = ew * -math.sin(t)
	elif eh <= 2.05e+16:
		tmp = ew * math.sin(t)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(atan(Float64(Float64(eh / ew) / t))))
	tmp = 0.0
	if (eh <= -0.52)
		tmp = t_1;
	elseif (eh <= -5.6e-229)
		tmp = Float64(ew * Float64(-sin(t)));
	elseif (eh <= 2.05e+16)
		tmp = Float64(ew * sin(t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(atan(((eh / ew) / t)));
	tmp = 0.0;
	if (eh <= -0.52)
		tmp = t_1;
	elseif (eh <= -5.6e-229)
		tmp = ew * -sin(t);
	elseif (eh <= 2.05e+16)
		tmp = ew * sin(t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -0.52], t$95$1, If[LessEqual[eh, -5.6e-229], N[(ew * (-N[Sin[t], $MachinePrecision])), $MachinePrecision], If[LessEqual[eh, 2.05e+16], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -0.52000000000000002 or 2.05e16 < 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. Step-by-step derivation

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 55.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. add-sqr-sqrt 27.3%

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

      4. fabs-sqr 27.3%

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

      5. add-sqr-sqrt 27.6%

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

      6. associate-/r* 27.6%

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

   7. Applied egg-rr 27.6%

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

   8. Taylor expanded in t around 0 26.8%

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

      1. rem-cube-cbrt 27.3%

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

      2. *-commutative 27.3%

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

      3. associate-/r* 27.3%

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

   10. Applied egg-rr 27.3%

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

   if -0.52000000000000002 < eh < -5.5999999999999998e-229

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 41.5%

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

   7. Taylor expanded in ew around inf 23.2%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 22.2%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 44.3%

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

      3. pow2 44.3%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   10. Taylor expanded in ew around -inf 36.7%

       \[\leadsto \color{blue}{-1 \cdot \left(ew \cdot \sin t\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 36.7%

          \[\leadsto \color{blue}{-ew \cdot \sin t} \]

       2. distribute-rgt-neg-in 36.7%

          \[\leadsto \color{blue}{ew \cdot \left(-\sin t\right)} \]

   12. Simplified 36.7%

       \[\leadsto \color{blue}{ew \cdot \left(-\sin t\right)} \]

   if -5.5999999999999998e-229 < eh < 2.05e16

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 50.5%

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

   7. Taylor expanded in ew around inf 45.8%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -0.52:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \mathbf{elif}\;eh \leq -5.6 \cdot 10^{-229}:\\ \;\;\;\;ew \cdot \left(-\sin t\right)\\ \mathbf{elif}\;eh \leq 2.05 \cdot 10^{+16}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 24.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{t}\\ t_2 := ew \cdot \left(-\sin t\right)\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{eh \cdot t\_1}{\mathsf{hypot}\left(1, t\_1\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{{\left(t \cdot ew\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+177}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t)) (t_2 (* ew (- (sin t)))))
   (if (<= t -4.4e-130)
     t_2
     (if (<= t -1e-214)
       (/ (* eh t_1) (hypot 1.0 t_1))
       (if (<= t 3.5e+18)
         (sqrt (pow (* t ew) 2.0))
         (if (<= t 3.5e+177) t_2 (* ew (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = ew * -sin(t);
	double tmp;
	if (t <= -4.4e-130) {
		tmp = t_2;
	} else if (t <= -1e-214) {
		tmp = (eh * t_1) / hypot(1.0, t_1);
	} else if (t <= 3.5e+18) {
		tmp = sqrt(pow((t * ew), 2.0));
	} else if (t <= 3.5e+177) {
		tmp = t_2;
	} else {
		tmp = ew * sin(t);
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = ew * -Math.sin(t);
	double tmp;
	if (t <= -4.4e-130) {
		tmp = t_2;
	} else if (t <= -1e-214) {
		tmp = (eh * t_1) / Math.hypot(1.0, t_1);
	} else if (t <= 3.5e+18) {
		tmp = Math.sqrt(Math.pow((t * ew), 2.0));
	} else if (t <= 3.5e+177) {
		tmp = t_2;
	} else {
		tmp = ew * Math.sin(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	t_2 = ew * -math.sin(t)
	tmp = 0
	if t <= -4.4e-130:
		tmp = t_2
	elif t <= -1e-214:
		tmp = (eh * t_1) / math.hypot(1.0, t_1)
	elif t <= 3.5e+18:
		tmp = math.sqrt(math.pow((t * ew), 2.0))
	elif t <= 3.5e+177:
		tmp = t_2
	else:
		tmp = ew * math.sin(t)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	t_2 = Float64(ew * Float64(-sin(t)))
	tmp = 0.0
	if (t <= -4.4e-130)
		tmp = t_2;
	elseif (t <= -1e-214)
		tmp = Float64(Float64(eh * t_1) / hypot(1.0, t_1));
	elseif (t <= 3.5e+18)
		tmp = sqrt((Float64(t * ew) ^ 2.0));
	elseif (t <= 3.5e+177)
		tmp = t_2;
	else
		tmp = Float64(ew * sin(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	t_2 = ew * -sin(t);
	tmp = 0.0;
	if (t <= -4.4e-130)
		tmp = t_2;
	elseif (t <= -1e-214)
		tmp = (eh * t_1) / hypot(1.0, t_1);
	elseif (t <= 3.5e+18)
		tmp = sqrt(((t * ew) ^ 2.0));
	elseif (t <= 3.5e+177)
		tmp = t_2;
	else
		tmp = ew * sin(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(ew * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t, -4.4e-130], t$95$2, If[LessEqual[t, -1e-214], N[(N[(eh * t$95$1), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e+18], N[Sqrt[N[Power[N[(t * ew), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 3.5e+177], t$95$2, N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.3999999999999997e-130 or 3.5e18 < t < 3.49999999999999991e177

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 41.1%

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

   7. Taylor expanded in ew around inf 17.8%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 16.8%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 32.9%

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

      3. pow2 32.9%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   10. Taylor expanded in ew around -inf 34.1%

       \[\leadsto \color{blue}{-1 \cdot \left(ew \cdot \sin t\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 34.1%

          \[\leadsto \color{blue}{-ew \cdot \sin t} \]

       2. distribute-rgt-neg-in 34.1%

          \[\leadsto \color{blue}{ew \cdot \left(-\sin t\right)} \]

   12. Simplified 34.1%

       \[\leadsto \color{blue}{ew \cdot \left(-\sin t\right)} \]

   if -4.3999999999999997e-130 < t < -9.99999999999999913e-215

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 89.8%

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

      1. add-cube-cbrt 87.6%

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

      2. pow3 87.6%

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

      3. add-sqr-sqrt 36.2%

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

      4. fabs-sqr 36.2%

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

      5. add-sqr-sqrt 38.2%

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

      6. associate-/r* 38.2%

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

   7. Applied egg-rr 38.2%

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

   8. Taylor expanded in t around 0 38.2%

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

      1. rem-cube-cbrt 38.8%

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

      2. *-commutative 38.8%

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

      3. sin-atan 11.9%

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

      4. associate-*l/ 11.9%

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

      5. associate-/r* 12.7%

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

      6. hypot-1-def 23.0%

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

      7. associate-/r* 27.9%

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

   10. Applied egg-rr 27.9%

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

   if -9.99999999999999913e-215 < t < 3.5e18

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.7%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in ew around inf 27.3%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 26.5%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 30.3%

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

      3. pow2 30.3%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   10. Taylor expanded in t around 0 10.1%

       \[\leadsto \sqrt{\color{blue}{{ew}^{2} \cdot {t}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 10.1%

          \[\leadsto \sqrt{\color{blue}{\left(ew \cdot ew\right)} \cdot {t}^{2}} \]

       2. unpow2 10.1%

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

       3. swap-sqr 29.2%

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

       4. unpow2 29.2%

          \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot t\right)}^{2}}} \]

   12. Simplified 29.2%

       \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot t\right)}^{2}}} \]

   if 3.49999999999999991e177 < 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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.5%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 54.7%

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

   7. Taylor expanded in ew around inf 29.6%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-130}:\\ \;\;\;\;ew \cdot \left(-\sin t\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{eh \cdot \frac{\frac{eh}{ew}}{t}}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{t}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{{\left(t \cdot ew\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+177}:\\ \;\;\;\;ew \cdot \left(-\sin t\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 22.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-130} \lor \neg \left(t \leq 4400000000000\right) \land t \leq 7.5 \cdot 10^{+176}:\\ \;\;\;\;ew \cdot \left(-\sin t\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -4.4e-130) (and (not (<= t 4400000000000.0)) (<= t 7.5e+176)))
   (* ew (- (sin t)))
   (* ew (sin t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.4e-130) || (!(t <= 4400000000000.0) && (t <= 7.5e+176))) {
		tmp = ew * -sin(t);
	} else {
		tmp = ew * sin(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 <= (-4.4d-130)) .or. (.not. (t <= 4400000000000.0d0)) .and. (t <= 7.5d+176)) then
        tmp = ew * -sin(t)
    else
        tmp = ew * sin(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.4e-130) || (!(t <= 4400000000000.0) && (t <= 7.5e+176))) {
		tmp = ew * -Math.sin(t);
	} else {
		tmp = ew * Math.sin(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -4.4e-130) or (not (t <= 4400000000000.0) and (t <= 7.5e+176)):
		tmp = ew * -math.sin(t)
	else:
		tmp = ew * math.sin(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -4.4e-130) || (!(t <= 4400000000000.0) && (t <= 7.5e+176)))
		tmp = Float64(ew * Float64(-sin(t)));
	else
		tmp = Float64(ew * sin(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -4.4e-130) || (~((t <= 4400000000000.0)) && (t <= 7.5e+176)))
		tmp = ew * -sin(t);
	else
		tmp = ew * sin(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -4.4e-130], And[N[Not[LessEqual[t, 4400000000000.0]], $MachinePrecision], LessEqual[t, 7.5e+176]]], N[(ew * (-N[Sin[t], $MachinePrecision])), $MachinePrecision], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3999999999999997e-130 or 4.4e12 < t < 7.499999999999999e176

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 40.4%

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

   7. Taylor expanded in ew around inf 17.5%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 16.6%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      2. sqrt-unprod 32.4%

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

      3. pow2 32.4%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \sin t\right)}^{2}}} \]

   9. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]

   10. Taylor expanded in ew around -inf 33.6%

       \[\leadsto \color{blue}{-1 \cdot \left(ew \cdot \sin t\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 33.6%

          \[\leadsto \color{blue}{-ew \cdot \sin t} \]

       2. distribute-rgt-neg-in 33.6%

          \[\leadsto \color{blue}{ew \cdot \left(-\sin t\right)} \]

   12. Simplified 33.6%

       \[\leadsto \color{blue}{ew \cdot \left(-\sin t\right)} \]

   if -4.3999999999999997e-130 < t < 4.4e12 or 7.499999999999999e176 < t

   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. Step-by-step derivation

      1. fma-define 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.7%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in ew around inf 26.4%

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-130} \lor \neg \left(t \leq 4400000000000\right) \land t \leq 7.5 \cdot 10^{+176}:\\ \;\;\;\;ew \cdot \left(-\sin t\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 22.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ ew \cdot \sin t \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (* ew (sin t)))

C

double code(double eh, double ew, double t) {
	return ew * sin(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 = ew * sin(t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 97.8%

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

   2. pow3 97.7%

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

6. Applied egg-rr 48.2%

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

7. Taylor expanded in ew around inf 22.2%

   \[\leadsto \color{blue}{ew \cdot \sin t} \]
8. Add Preprocessing

Alternative 19: 10.6% accurate, 306.3× speedup

Math

\[\begin{array}{l} \\ t \cdot ew \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return t * ew;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = t * ew
end function

Java

public static double code(double eh, double ew, double t) {
	return t * ew;
}

Python

def code(eh, ew, t):
	return t * ew

Julia

function code(eh, ew, t)
	return Float64(t * ew)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 97.8%

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

   2. pow3 97.7%

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

6. Applied egg-rr 48.2%

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

7. Taylor expanded in ew around inf 22.2%

   \[\leadsto \color{blue}{ew \cdot \sin t} \]

8. Taylor expanded in t around 0 12.1%

   \[\leadsto \color{blue}{ew \cdot t} \]

9. Final simplification 12.1%

   \[\leadsto t \cdot ew \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))))))