Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.1s

Alternatives: 10

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

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

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(eh * Float64(-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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((1.0 / (Math.hypot(1.0, (Math.tan(t) * (eh / ew))) / (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(((1.0 / (math.hypot(1.0, (math.tan(t) * (eh / ew))) / (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(1.0 / Float64(hypot(1.0, Float64(tan(t) * Float64(eh / ew))) / Float64(ew * cos(t)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew))))))
end

MATLAB

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

Wolfram

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

Derivation

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. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

4. Applied egg-rr 98.7%

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

   1. rem-cube-cbrt 99.9%

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

   2. clear-num 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 98.4% accurate, 1.5× speedup

Math

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

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 * Float64(-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 99.9%

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

   1. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in eh around 0 97.7%

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

6. Final simplification 97.7%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

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

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(Float64(eh * sin(t)) * sin(atan(Float64(eh * Float64(tan(t) / Float64(-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[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

4. Applied egg-rr 98.7%

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

   1. rem-cube-cbrt 99.9%

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

   2. clear-num 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in eh around 0 97.7%

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

   1. mul-1-neg 97.7%

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

   2. +-commutative 97.7%

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

   3. *-commutative 97.7%

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

   4. mul-1-neg 97.7%

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

   5. associate-/l* 97.7%

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

   6. distribute-lft-neg-out 97.7%

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

   7. *-commutative 97.7%

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

9. Simplified 97.7%

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

10. Final simplification 97.7%

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

Alternative 5: 74.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.0011:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{-ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -0.6)
   (fabs (* ew (cos t)))
   (if (<= t 0.0011)
     (fabs (- ew (* (sin (atan (* eh (/ (tan t) (- ew))))) (* t eh))))
     (fabs (* (* eh (sin t)) (sin (atan (* t (/ eh (- ew))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = fabs((ew * cos(t)));
	} else if (t <= 0.0011) {
		tmp = fabs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	} else {
		tmp = fabs(((eh * sin(t)) * sin(atan((t * (eh / -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 <= (-0.6d0)) then
        tmp = abs((ew * cos(t)))
    else if (t <= 0.0011d0) then
        tmp = abs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))))
    else
        tmp = abs(((eh * sin(t)) * sin(atan((t * (eh / -ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else if (t <= 0.0011) {
		tmp = Math.abs((ew - (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) * (t * eh))));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((t * (eh / -ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= -0.6:
		tmp = math.fabs((ew * math.cos(t)))
	elif t <= 0.0011:
		tmp = math.fabs((ew - (math.sin(math.atan((eh * (math.tan(t) / -ew)))) * (t * eh))))
	else:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((t * (eh / -ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = abs(Float64(ew * cos(t)));
	elseif (t <= 0.0011)
		tmp = abs(Float64(ew - Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) * Float64(t * eh))));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(t * Float64(eh / Float64(-ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = abs((ew * cos(t)));
	elseif (t <= 0.0011)
		tmp = abs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	else
		tmp = abs(((eh * sin(t)) * sin(atan((t * (eh / -ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -0.6], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.0011], N[Abs[N[(ew - N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in ew around inf 59.1%

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

   if -0.599999999999999978 < t < 0.00110000000000000007

   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. add-cube-cbrt 98.7%

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

      2. pow3 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. rem-cube-cbrt 100.0%

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

      2. clear-num 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

      3. associate-*r* 97.0%

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

      4. mul-1-neg 97.0%

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

      5. associate-/l* 97.0%

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

      6. distribute-lft-neg-out 97.0%

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

      7. distribute-lft-neg-out 97.0%

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

      8. distribute-rgt-neg-in 97.0%

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

   9. Simplified 97.0%

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

   if 0.00110000000000000007 < t

   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 ew around inf 92.3%

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

      1. +-commutative 92.3%

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

      2. mul-1-neg 92.3%

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

      3. unsub-neg 92.3%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in eh around inf 47.4%

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

      1. associate-*r/ 47.4%

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

      2. associate-*r* 47.4%

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

      3. neg-mul-1 47.4%

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

      4. mul-1-neg 47.4%

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

      5. *-commutative 47.4%

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

      6. associate-*r/ 47.4%

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

      7. *-commutative 47.4%

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

      8. distribute-rgt-neg-out 47.4%

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

   8. Simplified 47.4%

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

   9. Taylor expanded in t around 0 47.8%

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

       1. mul-1-neg 47.8%

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

       2. associate-/l* 47.8%

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

   11. Simplified 47.8%

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

   12. Taylor expanded in ew around 0 55.3%

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

       1. mul-1-neg 55.3%

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

       2. fabs-neg 55.3%

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

       3. associate-*r* 55.3%

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

       4. distribute-neg-frac2 55.3%

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

       5. *-commutative 55.3%

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

       6. associate-/l* 55.3%

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

   14. Simplified 55.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.0011:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{-ew}\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.00026:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot t\_1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (- ew))))
   (if (<= t -0.6)
     (fabs (* ew (cos t)))
     (if (<= t 0.00026)
       (fabs (+ ew (* eh (* t (sin (atan (* (tan t) t_1)))))))
       (fabs (* (* eh (sin t)) (sin (atan (* t t_1)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / -ew;
	double tmp;
	if (t <= -0.6) {
		tmp = fabs((ew * cos(t)));
	} else if (t <= 0.00026) {
		tmp = fabs((ew + (eh * (t * sin(atan((tan(t) * t_1)))))));
	} else {
		tmp = fabs(((eh * sin(t)) * sin(atan((t * t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = eh / -ew
    if (t <= (-0.6d0)) then
        tmp = abs((ew * cos(t)))
    else if (t <= 0.00026d0) then
        tmp = abs((ew + (eh * (t * sin(atan((tan(t) * t_1)))))))
    else
        tmp = abs(((eh * sin(t)) * sin(atan((t * t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / -ew;
	double tmp;
	if (t <= -0.6) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else if (t <= 0.00026) {
		tmp = Math.abs((ew + (eh * (t * Math.sin(Math.atan((Math.tan(t) * t_1)))))));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((t * t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / -ew
	tmp = 0
	if t <= -0.6:
		tmp = math.fabs((ew * math.cos(t)))
	elif t <= 0.00026:
		tmp = math.fabs((ew + (eh * (t * math.sin(math.atan((math.tan(t) * t_1)))))))
	else:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((t * t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(-ew))
	tmp = 0.0
	if (t <= -0.6)
		tmp = abs(Float64(ew * cos(t)));
	elseif (t <= 0.00026)
		tmp = abs(Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(tan(t) * t_1)))))));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(t * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / -ew;
	tmp = 0.0;
	if (t <= -0.6)
		tmp = abs((ew * cos(t)));
	elseif (t <= 0.00026)
		tmp = abs((ew + (eh * (t * sin(atan((tan(t) * t_1)))))));
	else
		tmp = abs(((eh * sin(t)) * sin(atan((t * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / (-ew)), $MachinePrecision]}, If[LessEqual[t, -0.6], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.00026], N[Abs[N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in ew around inf 59.1%

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

   if -0.599999999999999978 < t < 2.59999999999999977e-4

   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. Step-by-step derivation

      1. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 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. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. cos-atan 100.0%

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

      4. un-div-inv 100.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 98.6%

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

      7. sqr-neg 98.6%

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

      8. sqrt-unprod 98.6%

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

      9. add-sqr-sqrt 98.6%

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

      10. hypot-1-def 98.6%

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

      11. associate-*r/ 98.6%

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

      12. *-commutative 98.6%

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

      13. associate-/l* 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t around 0 97.0%

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

      1. mul-1-neg 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*r/ 97.0%

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

      4. *-commutative 97.0%

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

      5. distribute-rgt-neg-out 97.0%

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

   9. Simplified 97.0%

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

   if 2.59999999999999977e-4 < t

   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 ew around inf 92.3%

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

      1. +-commutative 92.3%

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

      2. mul-1-neg 92.3%

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

      3. unsub-neg 92.3%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in eh around inf 47.4%

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

      1. associate-*r/ 47.4%

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

      2. associate-*r* 47.4%

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

      3. neg-mul-1 47.4%

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

      4. mul-1-neg 47.4%

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

      5. *-commutative 47.4%

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

      6. associate-*r/ 47.4%

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

      7. *-commutative 47.4%

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

      8. distribute-rgt-neg-out 47.4%

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

   8. Simplified 47.4%

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

   9. Taylor expanded in t around 0 47.8%

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

       1. mul-1-neg 47.8%

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

       2. associate-/l* 47.8%

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

   11. Simplified 47.8%

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

   12. Taylor expanded in ew around 0 55.3%

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

       1. mul-1-neg 55.3%

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

       2. fabs-neg 55.3%

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

       3. associate-*r* 55.3%

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

       4. distribute-neg-frac2 55.3%

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

       5. *-commutative 55.3%

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

       6. associate-/l* 55.3%

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

   14. Simplified 55.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.00026:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7.5 \cdot 10^{-141} \lor \neg \left(ew \leq 1.7 \cdot 10^{-146}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{-ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7.5e-141) (not (<= ew 1.7e-146)))
   (fabs (* ew (cos t)))
   (fabs (* (* eh (sin t)) (sin (atan (* t (/ eh (- ew)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.5e-141) || !(ew <= 1.7e-146)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((eh * sin(t)) * sin(atan((t * (eh / -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 ((ew <= (-7.5d-141)) .or. (.not. (ew <= 1.7d-146))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs(((eh * sin(t)) * sin(atan((t * (eh / -ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.5e-141) || !(ew <= 1.7e-146)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((t * (eh / -ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -7.5e-141) or not (ew <= 1.7e-146):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((t * (eh / -ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7.5e-141) || !(ew <= 1.7e-146))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(t * Float64(eh / Float64(-ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -7.5e-141) || ~((ew <= 1.7e-146)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((eh * sin(t)) * sin(atan((t * (eh / -ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7.5e-141], N[Not[LessEqual[ew, 1.7e-146]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.50000000000000046e-141 or 1.7e-146 < ew

   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. add-cube-cbrt 98.4%

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

      2. pow3 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in ew around inf 78.2%

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

   if -7.50000000000000046e-141 < ew < 1.7e-146

   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. Taylor expanded in ew around inf 82.1%

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

      1. +-commutative 82.1%

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

      2. mul-1-neg 82.1%

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

      3. unsub-neg 82.1%

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

   5. Simplified 82.0%

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

   6. Taylor expanded in eh around inf 56.7%

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

      1. associate-*r/ 56.7%

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

      2. associate-*r* 56.7%

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

      3. neg-mul-1 56.7%

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

      4. mul-1-neg 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-*r/ 56.7%

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

      7. *-commutative 56.7%

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

      8. distribute-rgt-neg-out 56.7%

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

   8. Simplified 56.7%

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

   9. Taylor expanded in t around 0 56.8%

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

       1. mul-1-neg 56.8%

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

       2. associate-/l* 56.8%

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

   11. Simplified 56.8%

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

   12. Taylor expanded in ew around 0 74.5%

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

       1. mul-1-neg 74.5%

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

       2. fabs-neg 74.5%

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

       3. associate-*r* 74.5%

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

       4. distribute-neg-frac2 74.5%

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

       5. *-commutative 74.5%

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

       6. associate-/l* 74.5%

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

   14. Simplified 74.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.5 \cdot 10^{-141} \lor \neg \left(ew \leq 1.7 \cdot 10^{-146}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.26 \cdot 10^{+237} \lor \neg \left(eh \leq 2.2 \cdot 10^{+157}\right):\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.26e+237) (not (<= eh 2.2e+157)))
   (fabs (* eh (* t (sin (atan (/ (* t eh) (- ew)))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.26e+237) || !(eh <= 2.2e+157)) {
		tmp = fabs((eh * (t * sin(atan(((t * eh) / -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 <= (-1.26d+237)) .or. (.not. (eh <= 2.2d+157))) then
        tmp = abs((eh * (t * sin(atan(((t * eh) / -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 <= -1.26e+237) || !(eh <= 2.2e+157)) {
		tmp = Math.abs((eh * (t * Math.sin(Math.atan(((t * eh) / -ew))))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.26e+237) or not (eh <= 2.2e+157):
		tmp = math.fabs((eh * (t * math.sin(math.atan(((t * eh) / -ew))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.26e+237) || !(eh <= 2.2e+157))
		tmp = abs(Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / 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 <= -1.26e+237) || ~((eh <= 2.2e+157)))
		tmp = abs((eh * (t * sin(atan(((t * eh) / -ew))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.26e+237], N[Not[LessEqual[eh, 2.2e+157]], $MachinePrecision]], N[Abs[N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $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 < -1.2599999999999999e237 or 2.2000000000000001e157 < eh

   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. Taylor expanded in ew around inf 69.6%

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

      1. +-commutative 69.6%

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

      2. mul-1-neg 69.6%

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

      3. unsub-neg 69.6%

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

   5. Simplified 69.2%

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

   6. Taylor expanded in eh around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. associate-*r* 61.2%

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

      3. neg-mul-1 61.2%

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

      4. mul-1-neg 61.2%

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

      5. *-commutative 61.2%

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

      6. associate-*r/ 61.2%

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

      7. *-commutative 61.2%

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

      8. distribute-rgt-neg-out 61.2%

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

   8. Simplified 61.2%

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

   9. Taylor expanded in t around 0 61.2%

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

       1. mul-1-neg 61.2%

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

       2. associate-/l* 61.2%

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

   11. Simplified 61.2%

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

   12. Taylor expanded in t around 0 59.2%

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

       1. associate-*r* 59.2%

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

       2. neg-mul-1 59.2%

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

       3. distribute-neg-frac2 59.2%

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

       4. *-commutative 59.2%

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

   14. Simplified 59.2%

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

   if -1.2599999999999999e237 < eh < 2.2000000000000001e157

   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. add-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in ew around inf 70.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.26 \cdot 10^{+237} \lor \neg \left(eh \leq 2.2 \cdot 10^{+157}\right):\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.4% 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 99.9%

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

   1. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in ew around inf 62.8%

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

Alternative 10: 41.9% 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 99.9%

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

   1. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in t around 0 44.1%

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

Reproduce

herbie shell --seed 2024163 
(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)))))))