Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-neg 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. add-sqr-sqrt 47.1%

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

   6. sqrt-unprod 76.6%

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

   7. sqr-neg 76.6%

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

   8. sqrt-unprod 52.2%

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

   9. add-sqr-sqrt 99.3%

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

   10. associate-*l* 99.3%

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

4. Applied egg-rr 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. add-sqr-sqrt 47.1%

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

   5. sqrt-unprod 76.6%

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

   6. sqr-neg 76.6%

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

   7. sqrt-unprod 52.2%

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

   8. add-sqr-sqrt 99.3%

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

   9. associate-*l* 99.3%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-neg 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. add-sqr-sqrt 47.1%

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

   6. sqrt-unprod 76.6%

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

   7. sqr-neg 76.6%

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

   8. sqrt-unprod 52.2%

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

   9. add-sqr-sqrt 99.3%

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

   10. associate-*l* 99.3%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 74.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.8 \cdot 10^{+88} \lor \neg \left(eh \leq 5.5 \cdot 10^{-40}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.8e+88) (not (<= eh 5.5e-40)))
   (fabs (* eh (* (sin t) (sin (atan (* eh (/ t (- ew))))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.8e+88) || !(eh <= 5.5e-40)) {
		tmp = fabs((eh * (sin(t) * sin(atan((eh * (t / -ew)))))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-3.8d+88)) .or. (.not. (eh <= 5.5d-40))) then
        tmp = abs((eh * (sin(t) * sin(atan((eh * (t / -ew)))))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.8e+88) || !(eh <= 5.5e-40)) {
		tmp = Math.abs((eh * (Math.sin(t) * Math.sin(Math.atan((eh * (t / -ew)))))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -3.8e+88) or not (eh <= 5.5e-40):
		tmp = math.fabs((eh * (math.sin(t) * math.sin(math.atan((eh * (t / -ew)))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.8e+88) || !(eh <= 5.5e-40))
		tmp = abs(Float64(eh * Float64(sin(t) * sin(atan(Float64(eh * Float64(t / Float64(-ew))))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -3.8e+88) || ~((eh <= 5.5e-40)))
		tmp = abs((eh * (sin(t) * sin(atan((eh * (t / -ew)))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.8e+88], N[Not[LessEqual[eh, 5.5e-40]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(t / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.7999999999999997e88 or 5.50000000000000002e-40 < eh

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf 99.7%

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

      1. associate-/l* 99.7%

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

      2. fma-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-*r* 99.7%

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

      3. neg-mul-1 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in eh around inf 77.1%

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

       1. associate-*r* 77.1%

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

       2. neg-mul-1 77.1%

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

       3. mul-1-neg 77.1%

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

       4. associate-/l* 77.1%

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

       5. distribute-lft-neg-out 77.1%

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

   11. Simplified 77.1%

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

   if -3.7999999999999997e88 < eh < 5.50000000000000002e-40

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. fma-neg 99.9%

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

      3. associate-*l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. add-sqr-sqrt 55.4%

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

      6. sqrt-unprod 95.9%

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

      7. sqr-neg 95.9%

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

      8. sqrt-unprod 44.0%

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

      9. add-sqr-sqrt 99.1%

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

      10. associate-*l* 99.1%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf 82.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.8 \cdot 10^{+88} \lor \neg \left(eh \leq 5.5 \cdot 10^{-40}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-200} \lor \neg \left(ew \leq 2.4 \cdot 10^{-49}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{\frac{ew}{\tan t}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.9e-200) (not (<= ew 2.4e-49)))
   (fabs (* ew (cos t)))
   (* (sin t) (* eh (sin (atan (/ eh (/ ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.9e-200) || !(ew <= 2.4e-49)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = sin(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 <= (-2.9d-200)) .or. (.not. (ew <= 2.4d-49))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = sin(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 <= -2.9e-200) || !(ew <= 2.4e-49)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.sin(t) * (eh * Math.sin(Math.atan((eh / (ew / Math.tan(t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.9e-200) or not (ew <= 2.4e-49):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.sin(t) * (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-200) || !(ew <= 2.4e-49))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = Float64(sin(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 <= -2.9e-200) || ~((ew <= 2.4e-49)))
		tmp = abs((ew * cos(t)));
	else
		tmp = sin(t) * (eh * sin(atan((eh / (ew / tan(t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.9e-200], N[Not[LessEqual[ew, 2.4e-49]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sin[t], $MachinePrecision] * 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 < -2.9e-200 or 2.39999999999999992e-49 < ew

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. fma-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. distribute-lft-neg-in 99.8%

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

      5. add-sqr-sqrt 47.7%

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

      6. sqrt-unprod 78.9%

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

      7. sqr-neg 78.9%

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

      8. sqrt-unprod 51.6%

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

      9. add-sqr-sqrt 99.4%

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

      10. associate-*l* 99.4%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf 75.7%

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

   if -2.9e-200 < ew < 2.39999999999999992e-49

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 58.2%

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

      2. pow2 58.2%

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

   4. Applied egg-rr 58.2%

      \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}}\right| \]

   5. Taylor expanded in ew around 0 47.7%

      \[\leadsto \left|{\left(\sqrt{\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)}}\right)}^{2}\right| \]
   6. Step-by-step derivation

      1. associate-*r* 47.7%

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

      2. *-commutative 47.7%

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

      3. associate-*r/ 47.7%

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

      4. *-commutative 47.7%

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

      5. associate-*r/ 47.7%

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

      6. *-commutative 47.7%

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

      7. associate-/l* 47.7%

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

   7. Simplified 47.7%

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)}}\right)}^{2}\right| \]
   8. Step-by-step derivation

      1. unpow2 47.7%

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

      2. fabs-sqr 47.7%

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

      3. add-sqr-sqrt 48.7%

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

      4. associate-*r* 48.7%

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

      5. clear-num 48.7%

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

      6. un-div-inv 48.7%

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

   9. Applied egg-rr 48.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-200} \lor \neg \left(ew \leq 2.4 \cdot 10^{-49}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{\frac{ew}{\tan t}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -7.5 \cdot 10^{+178} \lor \neg \left(eh \leq 4 \cdot 10^{+198}\right):\\ \;\;\;\;\left|\left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -7.5e+178) (not (<= eh 4e+198)))
   (fabs (* (* t eh) (sin (atan (* eh (/ (tan t) ew))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.5e+178) || !(eh <= 4e+198)) {
		tmp = fabs(((t * eh) * sin(atan((eh * (tan(t) / ew))))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-7.5d+178)) .or. (.not. (eh <= 4d+198))) then
        tmp = abs(((t * eh) * sin(atan((eh * (tan(t) / ew))))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.5e+178) || !(eh <= 4e+198)) {
		tmp = Math.abs(((t * eh) * Math.sin(Math.atan((eh * (Math.tan(t) / ew))))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -7.5e+178) or not (eh <= 4e+198):
		tmp = math.fabs(((t * eh) * math.sin(math.atan((eh * (math.tan(t) / ew))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -7.5e+178) || !(eh <= 4e+198))
		tmp = abs(Float64(Float64(t * eh) * sin(atan(Float64(eh * Float64(tan(t) / ew))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -7.5e+178) || ~((eh <= 4e+198)))
		tmp = abs(((t * eh) * sin(atan((eh * (tan(t) / ew))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -7.5e+178], N[Not[LessEqual[eh, 4e+198]], $MachinePrecision]], N[Abs[N[(N[(t * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.4999999999999995e178 or 4.00000000000000007e198 < eh

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 56.7%

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

      2. pow2 56.7%

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

   4. Applied egg-rr 56.7%

      \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}}\right| \]

   5. Taylor expanded in ew around 0 49.9%

      \[\leadsto \left|{\left(\sqrt{\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)}}\right)}^{2}\right| \]
   6. Step-by-step derivation

      1. associate-*r* 49.9%

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

      2. *-commutative 49.9%

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

      3. associate-*r/ 49.9%

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

      4. *-commutative 49.9%

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

      5. associate-*r/ 49.9%

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

      6. *-commutative 49.9%

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

      7. associate-/l* 49.9%

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

   7. Simplified 49.9%

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

   8. Taylor expanded in t around 0 38.6%

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

      1. associate-*r* 38.6%

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

      2. associate-/l* 38.6%

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

   10. Simplified 38.6%

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

   if -7.4999999999999995e178 < eh < 4.00000000000000007e198

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. fma-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. distribute-lft-neg-in 99.8%

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

      5. add-sqr-sqrt 47.5%

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

      6. sqrt-unprod 90.3%

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

      7. sqr-neg 90.3%

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

      8. sqrt-unprod 51.8%

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

      9. add-sqr-sqrt 99.3%

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

      10. associate-*l* 99.3%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf 70.2%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.5 \cdot 10^{+178} \lor \neg \left(eh \leq 4 \cdot 10^{+198}\right):\\ \;\;\;\;\left|\left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-neg 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. add-sqr-sqrt 47.1%

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

   6. sqrt-unprod 76.6%

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

   7. sqr-neg 76.6%

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

   8. sqrt-unprod 52.2%

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

   9. add-sqr-sqrt 99.3%

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

   10. associate-*l* 99.3%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 60.6%

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

Alternative 9: 49.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.035 \lor \neg \left(t \leq 4.2 \cdot 10^{-7}\right):\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.035) (not (<= t 4.2e-7))) (* ew (cos t)) (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.035) || !(t <= 4.2e-7)) {
		tmp = ew * cos(t);
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.035d0)) .or. (.not. (t <= 4.2d-7))) then
        tmp = ew * cos(t)
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.035) || !(t <= 4.2e-7)) {
		tmp = ew * Math.cos(t);
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.035) or not (t <= 4.2e-7):
		tmp = ew * math.cos(t)
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.035) || !(t <= 4.2e-7))
		tmp = Float64(ew * cos(t));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.035) || ~((t <= 4.2e-7)))
		tmp = ew * cos(t);
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.035], N[Not[LessEqual[t, 4.2e-7]], $MachinePrecision]], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.035000000000000003 or 4.2e-7 < t

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. fma-neg 99.6%

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

      3. associate-*l* 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. add-sqr-sqrt 41.7%

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

      6. sqrt-unprod 69.4%

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

      7. sqr-neg 69.4%

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

      8. sqrt-unprod 57.6%

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

      9. add-sqr-sqrt 99.2%

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

      10. associate-*l* 99.2%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ew around inf 44.3%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 25.7%

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

      2. fabs-sqr 25.7%

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

      3. add-sqr-sqrt 26.7%

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

      4. *-commutative 26.7%

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

   7. Applied egg-rr 26.7%

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

   if -0.035000000000000003 < t < 4.2e-7

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-neg 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. add-sqr-sqrt 52.7%

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

      6. sqrt-unprod 84.1%

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

      7. sqr-neg 84.1%

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

      8. sqrt-unprod 46.6%

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

      9. add-sqr-sqrt 99.5%

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

      10. associate-*l* 99.5%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0 77.0%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.035 \lor \neg \left(t \leq 4.2 \cdot 10^{-7}\right):\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.8% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-neg 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. add-sqr-sqrt 47.1%

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

   6. sqrt-unprod 76.6%

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

   7. sqr-neg 76.6%

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

   8. sqrt-unprod 52.2%

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

   9. add-sqr-sqrt 99.3%

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

   10. associate-*l* 99.3%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 43.6%

   \[\leadsto \left|\color{blue}{ew}\right| \]
6. Add Preprocessing

Alternative 11: 22.3% accurate, 921.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-neg 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. add-sqr-sqrt 47.1%

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

   6. sqrt-unprod 76.6%

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

   7. sqr-neg 76.6%

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

   8. sqrt-unprod 52.2%

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

   9. add-sqr-sqrt 99.3%

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

   10. associate-*l* 99.3%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 43.6%

   \[\leadsto \left|\color{blue}{ew}\right| \]
6. Step-by-step derivation

   1. add-cbrt-cube 17.2%

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

   2. pow1/3 16.4%

      \[\leadsto \color{blue}{{\left(\left(\left|ew\right| \cdot \left|ew\right|\right) \cdot \left|ew\right|\right)}^{0.3333333333333333}} \]

   3. pow3 16.4%

      \[\leadsto {\color{blue}{\left({\left(\left|ew\right|\right)}^{3}\right)}}^{0.3333333333333333} \]

7. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{{\left({\left(\left|ew\right|\right)}^{3}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 17.2%

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

   2. rem-cbrt-cube 43.6%

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

   3. add-sqr-sqrt 22.7%

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

   4. fabs-sqr 22.7%

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

   5. *-un-lft-identity 22.7%

      \[\leadsto \color{blue}{1 \cdot \left(\sqrt{ew} \cdot \sqrt{ew}\right)} \]

   6. add-sqr-sqrt 23.5%

      \[\leadsto 1 \cdot \color{blue}{ew} \]

9. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{1 \cdot ew} \]

10. Final simplification 23.5%

    \[\leadsto ew \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))