Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.5s

Alternatives: 12

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/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. hypot-1-def 99.8%

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

   4. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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%

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

   5. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 98.9%

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

4. Final simplification 98.9%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. hypot-1-def 99.8%

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

   4. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 98.3%

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

Alternative 6: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.7 \cdot 10^{+79} \lor \neg \left(ew \leq 4.1 \cdot 10^{+128}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.7e+79) (not (<= ew 4.1e+128)))
   (fabs (* ew (sin t)))
   (fabs (* (cos t) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.7e+79) || !(ew <= 4.1e+128)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.7d+79)) .or. (.not. (ew <= 4.1d+128))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.7e+79) || !(ew <= 4.1e+128)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.7e+79) or not (ew <= 4.1e+128):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.7e+79) || !(ew <= 4.1e+128))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.7e+79) || ~((ew <= 4.1e+128)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.7e+79], N[Not[LessEqual[ew, 4.1e+128]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.70000000000000016e79 or 4.10000000000000012e128 < 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-sqr-sqrt 54.6%

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

      2. fabs-sqr 54.6%

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

      3. add-sqr-sqrt 55.3%

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

      4. add-cbrt-cube 19.0%

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

   6. Applied egg-rr 19.0%

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

   7. Taylor expanded in ew around inf 4.0%

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

      1. cube-prod 16.8%

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

   9. Simplified 16.8%

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

       1. rem-cbrt-cube 46.0%

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

       2. add-sqr-sqrt 45.4%

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

       3. sqrt-unprod 36.0%

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

       4. pow2 36.0%

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

   11. Applied egg-rr 36.0%

       \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 36.0%

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

       2. rem-sqrt-square 74.1%

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

   13. Simplified 74.1%

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

   if -1.70000000000000016e79 < ew < 4.10000000000000012e128

   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 79.2%

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

      1. associate-*r* 79.2%

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

      2. *-commutative 79.2%

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

      3. associate-*r* 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.7 \cdot 10^{+79} \lor \neg \left(ew \leq 4.1 \cdot 10^{+128}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+79} \lor \neg \left(ew \leq 1.06 \cdot 10^{+96}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.9e+79) (not (<= ew 1.06e+96)))
   (fabs (* ew (sin t)))
   (fabs (* eh (sin (atan (/ eh (* ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.9e+79) || !(ew <= 1.06e+96)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((eh * sin(atan((eh / (ew * tan(t)))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-2.9d+79)) .or. (.not. (ew <= 1.06d+96))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.9e+79) || !(ew <= 1.06e+96)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.9e+79) or not (ew <= 1.06e+96):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.9e+79) || !(ew <= 1.06e+96))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2.9e+79) || ~((ew <= 1.06e+96)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.9e+79], N[Not[LessEqual[ew, 1.06e+96]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.89999999999999992e79 or 1.06e96 < ew

   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. Step-by-step derivation

      1. add-sqr-sqrt 53.6%

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

      2. fabs-sqr 53.6%

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

      3. add-sqr-sqrt 54.3%

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

      4. add-cbrt-cube 17.7%

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

   6. Applied egg-rr 17.7%

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

   7. Taylor expanded in ew around inf 3.8%

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

      1. cube-prod 15.6%

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

   9. Simplified 15.6%

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

       1. rem-cbrt-cube 42.6%

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

       2. add-sqr-sqrt 41.9%

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

       3. sqrt-unprod 36.8%

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

       4. pow2 36.8%

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

   11. Applied egg-rr 36.8%

       \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 36.8%

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

       2. rem-sqrt-square 71.8%

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

   13. Simplified 71.8%

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

   if -2.89999999999999992e79 < ew < 1.06e96

   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 54.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+79} \lor \neg \left(ew \leq 1.06 \cdot 10^{+96}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-30} \lor \neg \left(t \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -4.4e-30) (not (<= t 1.4e-11)))
   (fabs (* ew (sin t)))
   (fabs (* eh (sin (atan (/ eh (* ew t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.4e-30) || !(t <= 1.4e-11)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((eh * sin(atan((eh / (ew * t))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.4d-30)) .or. (.not. (t <= 1.4d-11))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((eh * sin(atan((eh / (ew * t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.4e-30) || !(t <= 1.4e-11)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -4.4e-30) or not (t <= 1.4e-11):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -4.4e-30) || !(t <= 1.4e-11))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -4.4e-30) || ~((t <= 1.4e-11)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((eh * sin(atan((eh / (ew * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -4.4e-30], N[Not[LessEqual[t, 1.4e-11]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.39999999999999967e-30 or 1.4e-11 < t

   1. Initial program 99.7%

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

      1. 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-sqr-sqrt 49.1%

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

      2. fabs-sqr 49.1%

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

      3. add-sqr-sqrt 50.0%

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

      4. add-cbrt-cube 15.3%

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

   6. Applied egg-rr 15.3%

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

   7. Taylor expanded in ew around inf 8.0%

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

      1. cube-prod 8.0%

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

   9. Simplified 8.0%

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

       1. rem-cbrt-cube 25.4%

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

       2. add-sqr-sqrt 24.5%

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

       3. sqrt-unprod 23.6%

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

       4. pow2 23.6%

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

   11. Applied egg-rr 23.6%

       \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 23.6%

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

       2. rem-sqrt-square 47.3%

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

   13. Simplified 47.3%

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

   if -4.39999999999999967e-30 < t < 1.4e-11

   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 72.4%

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

   6. Taylor expanded in t around 0 72.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-30} \lor \neg \left(t \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $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. add-sqr-sqrt 46.5%

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

   2. fabs-sqr 46.5%

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

   3. add-sqr-sqrt 47.6%

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

   4. add-cbrt-cube 19.4%

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

6. Applied egg-rr 19.4%

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

7. Taylor expanded in ew around inf 7.4%

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

   1. cube-prod 11.4%

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

9. Simplified 11.4%

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

    1. rem-cbrt-cube 22.4%

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

    2. add-sqr-sqrt 21.5%

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

    3. sqrt-unprod 24.6%

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

    4. pow2 24.6%

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

11. Applied egg-rr 24.6%

    \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \sin t\right)}^{2}}} \]
12. Step-by-step derivation

    1. unpow2 24.6%

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

    2. rem-sqrt-square 39.5%

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

13. Simplified 39.5%

    \[\leadsto \color{blue}{\left|ew \cdot \sin t\right|} \]
14. Add Preprocessing

Alternative 10: 28.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \lor \neg \left(t \leq 0.0042\right):\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -8.0) (not (<= t 0.0042))) (* ew (sin t)) (fabs (* ew t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -8.0) || !(t <= 0.0042)) {
		tmp = ew * sin(t);
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-8.0d0)) .or. (.not. (t <= 0.0042d0))) then
        tmp = ew * sin(t)
    else
        tmp = abs((ew * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -8.0) || !(t <= 0.0042)) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = Math.abs((ew * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -8.0) or not (t <= 0.0042):
		tmp = ew * math.sin(t)
	else:
		tmp = math.fabs((ew * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -8.0) || !(t <= 0.0042))
		tmp = Float64(ew * sin(t));
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -8.0) || ~((t <= 0.0042)))
		tmp = ew * sin(t);
	else
		tmp = abs((ew * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -8.0], N[Not[LessEqual[t, 0.0042]], $MachinePrecision]], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8 or 0.00419999999999999974 < 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 99.7%

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

      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-sqr-sqrt 49.5%

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

      2. fabs-sqr 49.5%

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

      3. add-sqr-sqrt 50.4%

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

      4. add-cbrt-cube 16.1%

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

   6. Applied egg-rr 16.1%

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

   7. Taylor expanded in ew around inf 8.4%

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

      1. cube-prod 8.4%

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

   9. Simplified 8.4%

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

       1. rem-cbrt-cube 24.7%

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

       2. *-commutative 24.7%

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

   11. Applied egg-rr 24.7%

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

   if -8 < t < 0.00419999999999999974

   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-sqr-sqrt 43.8%

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

      2. fabs-sqr 43.8%

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

      3. add-sqr-sqrt 45.1%

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

      4. add-cbrt-cube 22.2%

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

   6. Applied egg-rr 22.2%

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

   7. Taylor expanded in ew around inf 6.5%

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

      1. cube-prod 14.1%

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

   9. Simplified 14.1%

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

   10. Taylor expanded in t around 0 20.4%

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

       1. add-sqr-sqrt 19.4%

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

       2. sqrt-unprod 24.8%

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

       3. pow2 24.8%

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

       4. *-commutative 24.8%

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

   12. Applied egg-rr 24.8%

       \[\leadsto \color{blue}{\sqrt{{\left(t \cdot ew\right)}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 24.8%

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

       2. rem-sqrt-square 33.0%

          \[\leadsto \color{blue}{\left|t \cdot ew\right|} \]

   14. Simplified 33.0%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \lor \neg \left(t \leq 0.0042\right):\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 18.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * 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-sqr-sqrt 46.5%

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

   2. fabs-sqr 46.5%

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

   3. add-sqr-sqrt 47.6%

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

   4. add-cbrt-cube 19.4%

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

6. Applied egg-rr 19.4%

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

7. Taylor expanded in ew around inf 7.4%

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

   1. cube-prod 11.4%

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

9. Simplified 11.4%

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

10. Taylor expanded in t around 0 12.7%

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

    1. add-sqr-sqrt 11.9%

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

    2. sqrt-unprod 16.1%

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

    3. pow2 16.1%

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

    4. *-commutative 16.1%

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

12. Applied egg-rr 16.1%

    \[\leadsto \color{blue}{\sqrt{{\left(t \cdot ew\right)}^{2}}} \]
13. Step-by-step derivation

    1. unpow2 16.1%

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

    2. rem-sqrt-square 20.6%

       \[\leadsto \color{blue}{\left|t \cdot ew\right|} \]

14. Simplified 20.6%

    \[\leadsto \color{blue}{\left|t \cdot ew\right|} \]

15. Final simplification 20.6%

    \[\leadsto \left|ew \cdot t\right| \]
16. Add Preprocessing

Alternative 12: 10.1% 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-sqr-sqrt 46.5%

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

   2. fabs-sqr 46.5%

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

   3. add-sqr-sqrt 47.6%

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

   4. add-cbrt-cube 19.4%

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

6. Applied egg-rr 19.4%

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

7. Taylor expanded in ew around inf 7.4%

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

   1. cube-prod 11.4%

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

9. Simplified 11.4%

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

10. Taylor expanded in t around 0 12.7%

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

Reproduce

herbie shell --seed 2024185 
(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))))))))