Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 22.5s

Alternatives: 19

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

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)) + (sin(atan(t_1)) * (eh * cos(t)))));
}

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

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)) + (math.sin(math.atan(t_1)) * (eh * math.cos(t)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in eh around 0 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 98.5%

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

   1. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 4: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in eh around 0 97.8%

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

Alternative 5: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ t_2 := \tan^{-1} t\_1\\ t_3 := \sin t\_2\\ t_4 := eh \cdot t\_3\\ \mathbf{if}\;t \leq -200000000000:\\ \;\;\;\;\mathsf{fma}\left(eh, \cos t \cdot t\_3, ew \cdot \sin t\right)\\ \mathbf{elif}\;t \leq 64000000:\\ \;\;\;\;\left|t\_4 + \frac{ew \cdot t}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+88}:\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot t\_4\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t)))
        (t_2 (atan t_1))
        (t_3 (sin t_2))
        (t_4 (* eh t_3)))
   (if (<= t -200000000000.0)
     (fma eh (* (cos t) t_3) (* ew (sin t)))
     (if (<= t 64000000.0)
       (fabs (+ t_4 (/ (* ew t) (hypot 1.0 t_1))))
       (if (<= t 1.02e+88)
         (fabs (* (sin t) (* ew (cos t_2))))
         (fabs (* (cos t) t_4)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	double t_2 = atan(t_1);
	double t_3 = sin(t_2);
	double t_4 = eh * t_3;
	double tmp;
	if (t <= -200000000000.0) {
		tmp = fma(eh, (cos(t) * t_3), (ew * sin(t)));
	} else if (t <= 64000000.0) {
		tmp = fabs((t_4 + ((ew * t) / hypot(1.0, t_1))));
	} else if (t <= 1.02e+88) {
		tmp = fabs((sin(t) * (ew * cos(t_2))));
	} else {
		tmp = fabs((cos(t) * t_4));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	t_2 = atan(t_1)
	t_3 = sin(t_2)
	t_4 = Float64(eh * t_3)
	tmp = 0.0
	if (t <= -200000000000.0)
		tmp = fma(eh, Float64(cos(t) * t_3), Float64(ew * sin(t)));
	elseif (t <= 64000000.0)
		tmp = abs(Float64(t_4 + Float64(Float64(ew * t) / hypot(1.0, t_1))));
	elseif (t <= 1.02e+88)
		tmp = abs(Float64(sin(t) * Float64(ew * cos(t_2))));
	else
		tmp = abs(Float64(cos(t) * t_4));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(eh * t$95$3), $MachinePrecision]}, If[LessEqual[t, -200000000000.0], N[(eh * N[(N[Cos[t], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 64000000.0], N[Abs[N[(t$95$4 + N[(N[(ew * t), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.02e+88], N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(ew * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2e11

   1. Initial program 99.4%

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

      1. associate-*l* 99.4%

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

      2. fma-define 99.4%

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

      3. associate-/r* 99.4%

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

      4. associate-*l* 99.4%

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

      5. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 98.0%

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

   6. Applied egg-rr 70.7%

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

   7. Taylor expanded in eh around 0 69.4%

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

      1. fma-define 69.4%

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

      2. associate-/r* 69.4%

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

   9. Simplified 69.4%

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

   if -2e11 < t < 6.4e7

   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. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

      3. associate-/r* 100.0%

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

      4. associate-*l* 100.0%

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

      5. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 97.7%

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

      1. fma-define 97.7%

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

      2. associate-/r* 97.7%

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

      3. associate-*r* 97.7%

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

      4. associate-/r* 97.7%

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

   7. Simplified 97.7%

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

      1. fma-undefine 97.7%

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

      2. associate-/r* 97.7%

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

      3. cos-atan 97.7%

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

      4. hypot-1-def 97.7%

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

      5. un-div-inv 97.7%

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

      6. associate-/r* 97.7%

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

   9. Applied egg-rr 97.7%

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

   if 6.4e7 < t < 1.01999999999999998e88

   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. associate-*l* 99.9%

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

      2. fma-define 99.9%

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

      3. associate-/r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 75.1%

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

      1. associate-/r* 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in ew around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-/r* 68.7%

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

   10. Simplified 68.7%

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

   if 1.01999999999999998e88 < t

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

      2. fma-define 99.5%

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

      3. associate-/r* 99.5%

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

      4. associate-*l* 99.5%

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

      5. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ew around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. *-commutative 69.8%

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

      3. associate-*l* 69.8%

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

      4. associate-/r* 69.8%

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

   7. Simplified 69.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -200000000000:\\ \;\;\;\;\mathsf{fma}\left(eh, \cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), ew \cdot \sin t\right)\\ \mathbf{elif}\;t \leq 64000000:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \frac{ew \cdot t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right|\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+88}:\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_2 := \left|\sin t \cdot \left(ew \cdot \cos t\_1\right)\right|\\ \mathbf{if}\;ew \leq -1 \cdot 10^{+127}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq -7.5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(eh, \cos t \cdot \sin t\_1, ew \cdot \sin t\right)\\ \mathbf{elif}\;ew \leq 1.9 \cdot 10^{+57}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t))))
        (t_2 (fabs (* (sin t) (* ew (cos t_1))))))
   (if (<= ew -1e+127)
     t_2
     (if (<= ew -7.5e-55)
       (fma eh (* (cos t) (sin t_1)) (* ew (sin t)))
       (if (<= ew 1.9e+57)
         (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	double t_2 = fabs((sin(t) * (ew * cos(t_1))));
	double tmp;
	if (ew <= -1e+127) {
		tmp = t_2;
	} else if (ew <= -7.5e-55) {
		tmp = fma(eh, (cos(t) * sin(t_1)), (ew * sin(t)));
	} else if (ew <= 1.9e+57) {
		tmp = fabs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	t_2 = abs(Float64(sin(t) * Float64(ew * cos(t_1))))
	tmp = 0.0
	if (ew <= -1e+127)
		tmp = t_2;
	elseif (ew <= -7.5e-55)
		tmp = fma(eh, Float64(cos(t) * sin(t_1)), Float64(ew * sin(t)));
	elseif (ew <= 1.9e+57)
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(ew * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1e+127], t$95$2, If[LessEqual[ew, -7.5e-55], N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 1.9e+57], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -9.99999999999999955e126 or 1.8999999999999999e57 < 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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 94.4%

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

      1. associate-/r* 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in ew around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. associate-/r* 77.1%

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

   10. Simplified 77.1%

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

   if -9.99999999999999955e126 < ew < -7.50000000000000023e-55

   1. Initial program 99.6%

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

      1. associate-*l* 99.5%

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

      2. fma-define 99.6%

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

      3. associate-/r* 99.6%

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

      4. associate-*l* 99.6%

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

      5. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in eh around 0 70.5%

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

      1. fma-define 70.6%

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

      2. associate-/r* 70.6%

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

   9. Simplified 70.6%

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

   if -7.50000000000000023e-55 < ew < 1.8999999999999999e57

   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. associate-*l* 99.8%

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

      2. fma-define 99.9%

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

      3. associate-/r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in ew around 0 88.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{+127}:\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{elif}\;ew \leq -7.5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(eh, \cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), ew \cdot \sin t\right)\\ \mathbf{elif}\;ew \leq 1.9 \cdot 10^{+57}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\\ t_2 := \left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{if}\;ew \leq -1.05 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq -7.5 \cdot 10^{-51}:\\ \;\;\;\;t\_1 + ew \cdot \sin t\\ \mathbf{elif}\;ew \leq 5.8 \cdot 10^{+52}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))
        (t_2 (fabs (* (sin t) (* ew (cos (atan (/ (/ eh ew) (tan t)))))))))
   (if (<= ew -1.05e+122)
     t_2
     (if (<= ew -7.5e-51)
       (+ t_1 (* ew (sin t)))
       (if (<= ew 5.8e+52) (fabs t_1) t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (cos(t) * sin(atan((eh / (ew * tan(t))))));
	double t_2 = fabs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))));
	double tmp;
	if (ew <= -1.05e+122) {
		tmp = t_2;
	} else if (ew <= -7.5e-51) {
		tmp = t_1 + (ew * sin(t));
	} else if (ew <= 5.8e+52) {
		tmp = fabs(t_1);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))
    t_2 = abs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))))
    if (ew <= (-1.05d+122)) then
        tmp = t_2
    else if (ew <= (-7.5d-51)) then
        tmp = t_1 + (ew * sin(t))
    else if (ew <= 5.8d+52) then
        tmp = abs(t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * Math.tan(t))))));
	double t_2 = Math.abs((Math.sin(t) * (ew * Math.cos(Math.atan(((eh / ew) / Math.tan(t)))))));
	double tmp;
	if (ew <= -1.05e+122) {
		tmp = t_2;
	} else if (ew <= -7.5e-51) {
		tmp = t_1 + (ew * Math.sin(t));
	} else if (ew <= 5.8e+52) {
		tmp = Math.abs(t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * (math.cos(t) * math.sin(math.atan((eh / (ew * math.tan(t))))))
	t_2 = math.fabs((math.sin(t) * (ew * math.cos(math.atan(((eh / ew) / math.tan(t)))))))
	tmp = 0
	if ew <= -1.05e+122:
		tmp = t_2
	elif ew <= -7.5e-51:
		tmp = t_1 + (ew * math.sin(t))
	elif ew <= 5.8e+52:
		tmp = math.fabs(t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	t_2 = abs(Float64(sin(t) * Float64(ew * cos(atan(Float64(Float64(eh / ew) / tan(t)))))))
	tmp = 0.0
	if (ew <= -1.05e+122)
		tmp = t_2;
	elseif (ew <= -7.5e-51)
		tmp = Float64(t_1 + Float64(ew * sin(t)));
	elseif (ew <= 5.8e+52)
		tmp = abs(t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * (cos(t) * sin(atan((eh / (ew * tan(t))))));
	t_2 = abs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))));
	tmp = 0.0;
	if (ew <= -1.05e+122)
		tmp = t_2;
	elseif (ew <= -7.5e-51)
		tmp = t_1 + (ew * sin(t));
	elseif (ew <= 5.8e+52)
		tmp = abs(t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(ew * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.05e+122], t$95$2, If[LessEqual[ew, -7.5e-51], N[(t$95$1 + N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 5.8e+52], N[Abs[t$95$1], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.05000000000000008e122 or 5.8e52 < 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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 94.4%

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

      1. associate-/r* 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in ew around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. associate-/r* 77.1%

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

   10. Simplified 77.1%

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

   if -1.05000000000000008e122 < ew < -7.49999999999999976e-51

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. fma-define 99.6%

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

      3. associate-/r* 99.6%

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

      4. associate-*l* 99.6%

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

      5. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in eh around 0 69.7%

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

   if -7.49999999999999976e-51 < ew < 5.8e52

   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. associate-*l* 99.8%

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

      2. fma-define 99.9%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in ew around 0 88.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.05 \cdot 10^{+122}:\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{elif}\;ew \leq -7.5 \cdot 10^{-51}:\\ \;\;\;\;eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \sin t\\ \mathbf{elif}\;ew \leq 5.8 \cdot 10^{+52}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{-25} \lor \neg \left(ew \leq 2.9 \cdot 10^{+53}\right):\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1e-25) (not (<= ew 2.9e+53)))
   (fabs (* (sin t) (* ew (cos (atan (/ (/ eh ew) (tan t)))))))
   (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1e-25) || !(ew <= 2.9e+53)) {
		tmp = fabs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs((eh * (cos(t) * 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 ((ew <= (-1d-25)) .or. (.not. (ew <= 2.9d+53))) then
        tmp = abs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))))
    else
        tmp = abs((eh * (cos(t) * 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 ((ew <= -1e-25) || !(ew <= 2.9e+53)) {
		tmp = Math.abs((Math.sin(t) * (ew * Math.cos(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1e-25) or not (ew <= 2.9e+53):
		tmp = math.fabs((math.sin(t) * (ew * math.cos(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1e-25) || !(ew <= 2.9e+53))
		tmp = abs(Float64(sin(t) * Float64(ew * cos(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1e-25) || ~((ew <= 2.9e+53)))
		tmp = abs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))));
	else
		tmp = abs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1e-25], N[Not[LessEqual[ew, 2.9e+53]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(ew * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.00000000000000004e-25 or 2.9000000000000002e53 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around 0 92.3%

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

      1. associate-/r* 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in ew around inf 71.5%

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

      1. associate-*r* 71.5%

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

      2. associate-/r* 71.5%

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

   10. Simplified 71.5%

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

   if -1.00000000000000004e-25 < ew < 2.9000000000000002e53

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 88.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{-25} \lor \neg \left(ew \leq 2.9 \cdot 10^{+53}\right):\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3.4 \cdot 10^{+75} \lor \neg \left(ew \leq 8.8 \cdot 10^{+55}\right):\\ \;\;\;\;{\left(\sqrt{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3.4e+75) (not (<= ew 8.8e+55)))
   (pow (sqrt (pow (cbrt (* ew (sin t))) 2.0)) 3.0)
   (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.4e+75) || !(ew <= 8.8e+55)) {
		tmp = pow(sqrt(pow(cbrt((ew * sin(t))), 2.0)), 3.0);
	} else {
		tmp = fabs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.4e+75) || !(ew <= 8.8e+55)) {
		tmp = Math.pow(Math.sqrt(Math.pow(Math.cbrt((ew * Math.sin(t))), 2.0)), 3.0);
	} else {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -3.4e+75) || !(ew <= 8.8e+55))
		tmp = sqrt((cbrt(Float64(ew * sin(t))) ^ 2.0)) ^ 3.0;
	else
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -3.4e+75], N[Not[LessEqual[ew, 8.8e+55]], $MachinePrecision]], N[Power[N[Sqrt[N[Power[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.40000000000000011e75 or 8.80000000000000042e55 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 57.5%

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

   7. Taylor expanded in ew around inf 46.7%

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

      1. add-sqr-sqrt 46.1%

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

      2. sqrt-unprod 74.1%

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

      3. pow2 74.1%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}}\right)}^{3} \]

   9. Applied egg-rr 74.1%

      \[\leadsto {\color{blue}{\left(\sqrt{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}\right)}}^{3} \]

   if -3.40000000000000011e75 < ew < 8.80000000000000042e55

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 83.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.4 \cdot 10^{+75} \lor \neg \left(ew \leq 8.8 \cdot 10^{+55}\right):\\ \;\;\;\;{\left(\sqrt{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{-25} \lor \neg \left(ew \leq 1.65 \cdot 10^{+28}\right):\\ \;\;\;\;{\left(\sqrt{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.35e-25) (not (<= ew 1.65e+28)))
   (pow (sqrt (pow (cbrt (* ew (sin t))) 2.0)) 3.0)
   (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.35e-25) || !(ew <= 1.65e+28)) {
		tmp = pow(sqrt(pow(cbrt((ew * sin(t))), 2.0)), 3.0);
	} else {
		tmp = fabs((eh * (cos(t) * sin(atan((eh / (ew * t)))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.35e-25) || !(ew <= 1.65e+28)) {
		tmp = Math.pow(Math.sqrt(Math.pow(Math.cbrt((ew * Math.sin(t))), 2.0)), 3.0);
	} else {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.35e-25) || !(ew <= 1.65e+28))
		tmp = sqrt((cbrt(Float64(ew * sin(t))) ^ 2.0)) ^ 3.0;
	else
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.35e-25], N[Not[LessEqual[ew, 1.65e+28]], $MachinePrecision]], N[Power[N[Sqrt[N[Power[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.35000000000000008e-25 or 1.65e28 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in ew around inf 45.3%

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

      1. add-sqr-sqrt 44.8%

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

      2. sqrt-unprod 69.6%

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

      3. pow2 69.6%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}}\right)}^{3} \]

   9. Applied egg-rr 69.6%

      \[\leadsto {\color{blue}{\left(\sqrt{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}\right)}}^{3} \]

   if -1.35000000000000008e-25 < ew < 1.65e28

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 88.7%

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

   6. Taylor expanded in t around 0 76.0%

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

      1. *-commutative 99.2%

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

   8. Simplified 76.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{-25} \lor \neg \left(ew \leq 1.65 \cdot 10^{+28}\right):\\ \;\;\;\;{\left(\sqrt{{\left(\sqrt[3]{ew \cdot \sin t}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -8.6 \cdot 10^{+214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -1.6 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{{t\_1}^{2}}\\ \mathbf{elif}\;ew \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -8.6e+214)
     t_1
     (if (<= ew -1.6e-25)
       (sqrt (pow t_1 2.0))
       (if (<= ew 2.1e+25)
         (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew t)))))))
         (* ew (sqrt (pow (sin t) 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -8.6e+214) {
		tmp = t_1;
	} else if (ew <= -1.6e-25) {
		tmp = sqrt(pow(t_1, 2.0));
	} else if (ew <= 2.1e+25) {
		tmp = fabs((eh * (cos(t) * 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) :: t_1
    real(8) :: tmp
    t_1 = ew * sin(t)
    if (ew <= (-8.6d+214)) then
        tmp = t_1
    else if (ew <= (-1.6d-25)) then
        tmp = sqrt((t_1 ** 2.0d0))
    else if (ew <= 2.1d+25) then
        tmp = abs((eh * (cos(t) * 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 t_1 = ew * Math.sin(t);
	double tmp;
	if (ew <= -8.6e+214) {
		tmp = t_1;
	} else if (ew <= -1.6e-25) {
		tmp = Math.sqrt(Math.pow(t_1, 2.0));
	} else if (ew <= 2.1e+25) {
		tmp = Math.abs((eh * (Math.cos(t) * 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):
	t_1 = ew * math.sin(t)
	tmp = 0
	if ew <= -8.6e+214:
		tmp = t_1
	elif ew <= -1.6e-25:
		tmp = math.sqrt(math.pow(t_1, 2.0))
	elif ew <= 2.1e+25:
		tmp = math.fabs((eh * (math.cos(t) * 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)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -8.6e+214)
		tmp = t_1;
	elseif (ew <= -1.6e-25)
		tmp = sqrt((t_1 ^ 2.0));
	elseif (ew <= 2.1e+25)
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = Float64(ew * sqrt((sin(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -8.6e+214)
		tmp = t_1;
	elseif (ew <= -1.6e-25)
		tmp = sqrt((t_1 ^ 2.0));
	elseif (ew <= 2.1e+25)
		tmp = abs((eh * (cos(t) * sin(atan((eh / (ew * t)))))));
	else
		tmp = ew * sqrt((sin(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -8.6e+214], t$95$1, If[LessEqual[ew, -1.6e-25], N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.1e+25], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * 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 4 regimes

2. if ew < -8.59999999999999966e214

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in ew around inf 68.4%

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

   if -8.59999999999999966e214 < ew < -1.6000000000000001e-25

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 60.3%

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

   7. Taylor expanded in ew around inf 39.7%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 52.9%

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

      3. pow2 52.9%

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

   9. Applied egg-rr 52.9%

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

   if -1.6000000000000001e-25 < ew < 2.0999999999999999e25

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 88.7%

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

   6. Taylor expanded in t around 0 76.0%

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

      1. *-commutative 99.2%

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

   8. Simplified 76.0%

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

   if 2.0999999999999999e25 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in ew around inf 43.0%

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

      1. add-sqr-sqrt 42.3%

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

      2. sqrt-unprod 60.4%

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

      3. pow2 60.4%

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

   9. Applied egg-rr 60.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -8.6 \cdot 10^{+214}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;ew \leq -1.6 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -2.05 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -7 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{{t\_1}^{2}}\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{+32}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -2.05e+215)
     t_1
     (if (<= ew -7e-26)
       (sqrt (pow t_1 2.0))
       (if (<= ew 2.9e+32)
         (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 t_1 = ew * sin(t);
	double tmp;
	if (ew <= -2.05e+215) {
		tmp = t_1;
	} else if (ew <= -7e-26) {
		tmp = sqrt(pow(t_1, 2.0));
	} else if (ew <= 2.9e+32) {
		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) :: t_1
    real(8) :: tmp
    t_1 = ew * sin(t)
    if (ew <= (-2.05d+215)) then
        tmp = t_1
    else if (ew <= (-7d-26)) then
        tmp = sqrt((t_1 ** 2.0d0))
    else if (ew <= 2.9d+32) 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 t_1 = ew * Math.sin(t);
	double tmp;
	if (ew <= -2.05e+215) {
		tmp = t_1;
	} else if (ew <= -7e-26) {
		tmp = Math.sqrt(Math.pow(t_1, 2.0));
	} else if (ew <= 2.9e+32) {
		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):
	t_1 = ew * math.sin(t)
	tmp = 0
	if ew <= -2.05e+215:
		tmp = t_1
	elif ew <= -7e-26:
		tmp = math.sqrt(math.pow(t_1, 2.0))
	elif ew <= 2.9e+32:
		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)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -2.05e+215)
		tmp = t_1;
	elseif (ew <= -7e-26)
		tmp = sqrt((t_1 ^ 2.0));
	elseif (ew <= 2.9e+32)
		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)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -2.05e+215)
		tmp = t_1;
	elseif (ew <= -7e-26)
		tmp = sqrt((t_1 ^ 2.0));
	elseif (ew <= 2.9e+32)
		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_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.05e+215], t$95$1, If[LessEqual[ew, -7e-26], N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.9e+32], 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 4 regimes

2. if ew < -2.0500000000000002e215

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in ew around inf 68.4%

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

   if -2.0500000000000002e215 < ew < -6.9999999999999997e-26

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 60.3%

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

   7. Taylor expanded in ew around inf 39.7%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 52.9%

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

      3. pow2 52.9%

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

   9. Applied egg-rr 52.9%

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

   if -6.9999999999999997e-26 < ew < 2.90000000000000003e32

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 88.7%

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

   6. Taylor expanded in t around 0 60.2%

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

   if 2.90000000000000003e32 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in ew around inf 43.0%

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

      1. add-sqr-sqrt 42.3%

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

      2. sqrt-unprod 60.4%

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

      3. pow2 60.4%

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

   9. Applied egg-rr 60.4%

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

Alternative 13: 60.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.2 \cdot 10^{-53}:\\ \;\;\;\;ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{elif}\;ew \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.2e-53)
   (+ (* ew (sin t)) (* eh (sin (atan (/ (/ eh ew) (tan t))))))
   (if (<= ew 1.65e+39)
     (fabs (* eh (* (cos t) (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 <= -1.2e-53) {
		tmp = (ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))));
	} else if (ew <= 1.65e+39) {
		tmp = fabs((eh * (cos(t) * 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 <= (-1.2d-53)) then
        tmp = (ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))
    else if (ew <= 1.65d+39) then
        tmp = abs((eh * (cos(t) * 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 <= -1.2e-53) {
		tmp = (ew * Math.sin(t)) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))));
	} else if (ew <= 1.65e+39) {
		tmp = Math.abs((eh * (Math.cos(t) * 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 <= -1.2e-53:
		tmp = (ew * math.sin(t)) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))
	elif ew <= 1.65e+39:
		tmp = math.fabs((eh * (math.cos(t) * 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 <= -1.2e-53)
		tmp = Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t))))));
	elseif (ew <= 1.65e+39)
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(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 <= -1.2e-53)
		tmp = (ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))));
	elseif (ew <= 1.65e+39)
		tmp = abs((eh * (cos(t) * 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, -1.2e-53], N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 1.65e+39], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * 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 < -1.20000000000000004e-53

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in eh around 0 66.3%

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

   8. Taylor expanded in t around 0 58.7%

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

      1. associate-/r* 91.1%

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

   10. Simplified 58.7%

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

   if -1.20000000000000004e-53 < ew < 1.6500000000000001e39

   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. associate-*l* 99.8%

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

      2. fma-define 99.9%

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

      3. associate-/r* 99.9%

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

      4. associate-*l* 99.9%

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

      5. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in ew around 0 89.0%

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

   6. Taylor expanded in t around 0 77.2%

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

      1. *-commutative 99.2%

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

   8. Simplified 77.2%

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

   if 1.6500000000000001e39 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in ew around inf 43.0%

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

      1. add-sqr-sqrt 42.3%

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

      2. sqrt-unprod 60.4%

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

      3. pow2 60.4%

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

   9. Applied egg-rr 60.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.2 \cdot 10^{-53}:\\ \;\;\;\;ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{elif}\;ew \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -2 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{{t\_1}^{2}}\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -2e+215)
     t_1
     (if (<= ew -5e-31)
       (sqrt (pow t_1 2.0))
       (if (<= ew 1.35e-5)
         (* eh (sin (atan (/ eh (* ew (tan t))))))
         (* ew (sqrt (pow (sin t) 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -2e+215) {
		tmp = t_1;
	} else if (ew <= -5e-31) {
		tmp = sqrt(pow(t_1, 2.0));
	} else if (ew <= 1.35e-5) {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = ew * sin(t)
    if (ew <= (-2d+215)) then
        tmp = t_1
    else if (ew <= (-5d-31)) then
        tmp = sqrt((t_1 ** 2.0d0))
    else if (ew <= 1.35d-5) then
        tmp = 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 t_1 = ew * Math.sin(t);
	double tmp;
	if (ew <= -2e+215) {
		tmp = t_1;
	} else if (ew <= -5e-31) {
		tmp = Math.sqrt(Math.pow(t_1, 2.0));
	} else if (ew <= 1.35e-5) {
		tmp = 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):
	t_1 = ew * math.sin(t)
	tmp = 0
	if ew <= -2e+215:
		tmp = t_1
	elif ew <= -5e-31:
		tmp = math.sqrt(math.pow(t_1, 2.0))
	elif ew <= 1.35e-5:
		tmp = 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)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -2e+215)
		tmp = t_1;
	elseif (ew <= -5e-31)
		tmp = sqrt((t_1 ^ 2.0));
	elseif (ew <= 1.35e-5)
		tmp = 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)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -2e+215)
		tmp = t_1;
	elseif (ew <= -5e-31)
		tmp = sqrt((t_1 ^ 2.0));
	elseif (ew <= 1.35e-5)
		tmp = 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_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2e+215], t$95$1, If[LessEqual[ew, -5e-31], N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.35e-5], N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $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 4 regimes

2. if ew < -1.99999999999999981e215

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in ew around inf 68.4%

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

   if -1.99999999999999981e215 < ew < -5e-31

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in ew around inf 39.0%

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

      1. add-sqr-sqrt 38.4%

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

      2. sqrt-unprod 52.0%

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

      3. pow2 52.0%

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

   9. Applied egg-rr 52.0%

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

   if -5e-31 < ew < 1.3499999999999999e-5

   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. associate-*l* 99.8%

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

      2. fma-define 99.9%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 60.3%

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

   7. Taylor expanded in t around 0 35.8%

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

      1. associate-/r* 35.8%

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

   9. Simplified 35.8%

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

       1. rem-cube-cbrt 36.5%

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

       2. associate-/r* 36.5%

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

       3. *-commutative 36.5%

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

   11. Applied egg-rr 36.5%

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

   if 1.3499999999999999e-5 < 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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.7%

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

   6. Applied egg-rr 52.8%

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

   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 41.3%

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

      2. sqrt-unprod 57.2%

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

      3. pow2 57.2%

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

   9. Applied egg-rr 57.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2 \cdot 10^{+215}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;ew \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \sin t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -2.4 \cdot 10^{+217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.3 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{{t\_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sqrt{{\sin t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -2.4e+217)
     t_1
     (if (<= ew 2.3e+207)
       (sqrt (pow t_1 2.0))
       (* ew (sqrt (pow (sin t) 2.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -2.4e+217) {
		tmp = t_1;
	} else if (ew <= 2.3e+207) {
		tmp = sqrt(pow(t_1, 2.0));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = ew * sin(t)
    if (ew <= (-2.4d+217)) then
        tmp = t_1
    else if (ew <= 2.3d+207) then
        tmp = sqrt((t_1 ** 2.0d0))
    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 t_1 = ew * Math.sin(t);
	double tmp;
	if (ew <= -2.4e+217) {
		tmp = t_1;
	} else if (ew <= 2.3e+207) {
		tmp = Math.sqrt(Math.pow(t_1, 2.0));
	} else {
		tmp = ew * Math.sqrt(Math.pow(Math.sin(t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	tmp = 0
	if ew <= -2.4e+217:
		tmp = t_1
	elif ew <= 2.3e+207:
		tmp = math.sqrt(math.pow(t_1, 2.0))
	else:
		tmp = ew * math.sqrt(math.pow(math.sin(t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -2.4e+217)
		tmp = t_1;
	elseif (ew <= 2.3e+207)
		tmp = sqrt((t_1 ^ 2.0));
	else
		tmp = Float64(ew * sqrt((sin(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -2.4e+217)
		tmp = t_1;
	elseif (ew <= 2.3e+207)
		tmp = sqrt((t_1 ^ 2.0));
	else
		tmp = ew * sqrt((sin(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.4e+217], t$95$1, If[LessEqual[ew, 2.3e+207], N[Sqrt[N[Power[t$95$1, 2.0], $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 < -2.3999999999999998e217

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in ew around inf 68.4%

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

   if -2.3999999999999998e217 < ew < 2.29999999999999995e207

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 57.0%

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

   7. Taylor expanded in ew around inf 20.1%

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

      1. add-sqr-sqrt 19.4%

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

      2. sqrt-unprod 28.3%

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

      3. pow2 28.3%

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

   9. Applied egg-rr 28.3%

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

   if 2.29999999999999995e207 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.7%

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

   6. Applied egg-rr 69.2%

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

   7. Taylor expanded in ew around inf 63.3%

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

      1. add-sqr-sqrt 63.0%

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

      2. sqrt-unprod 84.1%

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

      3. pow2 84.1%

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

   9. Applied egg-rr 84.1%

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

Alternative 16: 28.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -9.8 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 5.5 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{{t\_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \mathsf{fma}\left(ew, \sin t, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -9.8e+213)
     t_1
     (if (<= ew 5.5e+207)
       (sqrt (pow t_1 2.0))
       (+ -1.0 (fma ew (sin t) 1.0))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -9.8e+213) {
		tmp = t_1;
	} else if (ew <= 5.5e+207) {
		tmp = sqrt(pow(t_1, 2.0));
	} else {
		tmp = -1.0 + fma(ew, sin(t), 1.0);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -9.8e+213)
		tmp = t_1;
	elseif (ew <= 5.5e+207)
		tmp = sqrt((t_1 ^ 2.0));
	else
		tmp = Float64(-1.0 + fma(ew, sin(t), 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -9.8e+213], t$95$1, If[LessEqual[ew, 5.5e+207], N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], N[(-1.0 + N[(ew * N[Sin[t], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -9.79999999999999994e213

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in ew around inf 68.4%

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

   if -9.79999999999999994e213 < ew < 5.50000000000000036e207

   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. associate-*l* 99.8%

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

      2. fma-define 99.8%

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

      3. associate-/r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 57.0%

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

   7. Taylor expanded in ew around inf 20.1%

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

      1. add-sqr-sqrt 19.4%

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

      2. sqrt-unprod 28.3%

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

      3. pow2 28.3%

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

   9. Applied egg-rr 28.3%

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

   if 5.50000000000000036e207 < ew

   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. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

      3. associate-/r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.7%

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

   6. Applied egg-rr 69.2%

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

   7. Taylor expanded in ew around inf 62.0%

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

      1. rem-cube-cbrt 63.3%

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

      2. expm1-log1p-u 57.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \sin t\right)\right)} \]

      3. expm1-undefine 57.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(ew \cdot \sin t\right)} - 1} \]

   9. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(ew \cdot \sin t\right)} - 1} \]
   10. Step-by-step derivation

       1. sub-neg 57.1%

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

       2. metadata-eval 57.1%

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

       3. +-commutative 57.1%

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

       4. log1p-undefine 57.1%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + ew \cdot \sin t\right)}} \]

       5. rem-exp-log 63.3%

          \[\leadsto -1 + \color{blue}{\left(1 + ew \cdot \sin t\right)} \]

       6. +-commutative 63.3%

          \[\leadsto -1 + \color{blue}{\left(ew \cdot \sin t + 1\right)} \]

       7. fma-define 63.3%

          \[\leadsto -1 + \color{blue}{\mathsf{fma}\left(ew, \sin t, 1\right)} \]

   11. Simplified 63.3%

       \[\leadsto \color{blue}{-1 + \mathsf{fma}\left(ew, \sin t, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 25.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-40} \lor \neg \left(t \leq 7.5 \cdot 10^{-85}\right):\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(ew \cdot t\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.55e-40) (not (<= t 7.5e-85)))
   (* ew (sin t))
   (sqrt (pow (* ew t) 2.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.55e-40) || !(t <= 7.5e-85)) {
		tmp = ew * sin(t);
	} else {
		tmp = sqrt(pow((ew * 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 ((t <= (-1.55d-40)) .or. (.not. (t <= 7.5d-85))) then
        tmp = ew * sin(t)
    else
        tmp = sqrt(((ew * t) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.55e-40) || !(t <= 7.5e-85)) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = Math.sqrt(Math.pow((ew * t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -1.55e-40) or not (t <= 7.5e-85):
		tmp = ew * math.sin(t)
	else:
		tmp = math.sqrt(math.pow((ew * t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.55e-40) || !(t <= 7.5e-85))
		tmp = Float64(ew * sin(t));
	else
		tmp = sqrt((Float64(ew * t) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -1.55e-40) || ~((t <= 7.5e-85)))
		tmp = ew * sin(t);
	else
		tmp = sqrt(((ew * t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.55e-40], N[Not[LessEqual[t, 7.5e-85]], $MachinePrecision]], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(ew * t), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55000000000000005e-40 or 7.5000000000000003e-85 < t

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. fma-define 99.6%

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

      3. associate-/r* 99.6%

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

      4. associate-*l* 99.6%

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

      5. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 63.6%

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

   7. Taylor expanded in ew around inf 35.9%

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

   if -1.55000000000000005e-40 < t < 7.5000000000000003e-85

   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. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

      3. associate-/r* 100.0%

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

      4. associate-*l* 100.0%

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

      5. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   6. Applied egg-rr 53.9%

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

   7. Taylor expanded in ew around inf 18.5%

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

   8. Taylor expanded in t around 0 18.5%

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

      1. add-sqr-sqrt 17.8%

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

      2. sqrt-unprod 27.1%

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

      3. pow2 27.1%

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

   10. Applied egg-rr 27.1%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-40} \lor \neg \left(t \leq 7.5 \cdot 10^{-85}\right):\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(ew \cdot t\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 21.4% 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. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 97.8%

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

   2. pow3 97.8%

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

6. Applied egg-rr 59.4%

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

7. Taylor expanded in ew around inf 28.3%

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

Alternative 19: 10.2% accurate, 306.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(ew * t), $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. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 97.8%

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

   2. pow3 97.8%

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

6. Applied egg-rr 59.4%

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

7. Taylor expanded in ew around inf 28.3%

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

8. Taylor expanded in t around 0 14.4%

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

Reproduce

herbie shell --seed 2024123 
(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))))))))