Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.1s

Alternatives: 11

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $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. 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. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. add-sqr-sqrt 47.6%

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

   7. sqrt-unprod 91.0%

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

   8. sqr-neg 91.0%

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

   9. sqrt-unprod 52.2%

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

   10. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $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. 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. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. add-sqr-sqrt 47.6%

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

   7. sqrt-unprod 91.0%

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

   8. sqr-neg 91.0%

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

   9. sqrt-unprod 52.2%

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

   10. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

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(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[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $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. 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. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. add-sqr-sqrt 47.6%

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

   7. sqrt-unprod 91.0%

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

   8. sqr-neg 91.0%

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

   9. sqrt-unprod 52.2%

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

   10. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 98.1%

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

Alternative 4: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2 \cdot 10^{+122} \lor \neg \left(ew \leq 3.5 \cdot 10^{-37}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2e+122) (not (<= ew 3.5e-37)))
   (fabs (* (* ew (cos t)) (cos (atan (* eh (/ (tan t) (- ew)))))))
   (fabs (+ ew (* eh (* (sin t) (sin (atan (/ (* t eh) ew)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2e+122) || !(ew <= 3.5e-37)) {
		tmp = fabs(((ew * cos(t)) * cos(atan((eh * (tan(t) / -ew))))));
	} else {
		tmp = fabs((ew + (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 <= (-2d+122)) .or. (.not. (ew <= 3.5d-37))) then
        tmp = abs(((ew * cos(t)) * cos(atan((eh * (tan(t) / -ew))))))
    else
        tmp = abs((ew + (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 <= -2e+122) || !(ew <= 3.5e-37)) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(Math.atan((eh * (Math.tan(t) / -ew))))));
	} else {
		tmp = Math.abs((ew + (eh * (Math.sin(t) * Math.sin(Math.atan(((t * eh) / ew)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2e+122) or not (ew <= 3.5e-37):
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(math.atan((eh * (math.tan(t) / -ew))))))
	else:
		tmp = math.fabs((ew + (eh * (math.sin(t) * math.sin(math.atan(((t * eh) / ew)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2e+122) || !(ew <= 3.5e-37))
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	else
		tmp = abs(Float64(ew + Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(t * eh) / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2e+122) || ~((ew <= 3.5e-37)))
		tmp = abs(((ew * cos(t)) * cos(atan((eh * (tan(t) / -ew))))));
	else
		tmp = abs((ew + (eh * (sin(t) * sin(atan(((t * eh) / ew)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2e+122], N[Not[LessEqual[ew, 3.5e-37]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.00000000000000003e122 or 3.5000000000000001e-37 < 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. Taylor expanded in ew around inf 92.6%

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

      1. associate-*r* 92.6%

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

      2. *-commutative 92.6%

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

      3. mul-1-neg 92.6%

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

      4. associate-*r/ 92.6%

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

      5. distribute-rgt-neg-in 92.6%

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

      6. distribute-neg-frac2 92.6%

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

   5. Simplified 92.6%

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

   if -2.00000000000000003e122 < ew < 3.5000000000000001e-37

   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. cos-atan 99.8%

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

      3. un-div-inv 99.8%

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

      4. hypot-1-def 99.8%

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

      5. associate-/l* 99.8%

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

      6. add-sqr-sqrt 49.3%

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

      7. sqrt-unprod 96.4%

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

      8. sqr-neg 96.4%

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

      9. sqrt-unprod 50.5%

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

      10. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 98.3%

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

   6. Taylor expanded in t around 0 85.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2 \cdot 10^{+122} \lor \neg \left(ew \leq 3.5 \cdot 10^{-37}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3400 \lor \neg \left(t \leq 500\right):\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -3400.0) (not (<= t 500.0)))
   (fabs (* ew (cos (atan (/ (* eh (tan t)) ew)))))
   (fabs
    (+
     ew
     (* t (- (* t (* ew -0.5)) (* eh (sin (atan (/ (* t eh) (- ew)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -3400.0) || !(t <= 500.0)) {
		tmp = fabs((ew * cos(atan(((eh * tan(t)) / ew)))));
	} else {
		tmp = fabs((ew + (t * ((t * (ew * -0.5)) - (eh * 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 <= (-3400.0d0)) .or. (.not. (t <= 500.0d0))) then
        tmp = abs((ew * cos(atan(((eh * tan(t)) / ew)))))
    else
        tmp = abs((ew + (t * ((t * (ew * (-0.5d0))) - (eh * 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 <= -3400.0) || !(t <= 500.0)) {
		tmp = Math.abs((ew * Math.cos(Math.atan(((eh * Math.tan(t)) / ew)))));
	} else {
		tmp = Math.abs((ew + (t * ((t * (ew * -0.5)) - (eh * Math.sin(Math.atan(((t * eh) / -ew))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -3400.0) or not (t <= 500.0):
		tmp = math.fabs((ew * math.cos(math.atan(((eh * math.tan(t)) / ew)))))
	else:
		tmp = math.fabs((ew + (t * ((t * (ew * -0.5)) - (eh * math.sin(math.atan(((t * eh) / -ew))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -3400.0) || !(t <= 500.0))
		tmp = abs(Float64(ew * cos(atan(Float64(Float64(eh * tan(t)) / ew)))));
	else
		tmp = abs(Float64(ew + Float64(t * Float64(Float64(t * Float64(ew * -0.5)) - Float64(eh * sin(atan(Float64(Float64(t * eh) / Float64(-ew)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -3400.0) || ~((t <= 500.0)))
		tmp = abs((ew * cos(atan(((eh * tan(t)) / ew)))));
	else
		tmp = abs((ew + (t * ((t * (ew * -0.5)) - (eh * sin(atan(((t * eh) / -ew))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -3400.0], N[Not[LessEqual[t, 500.0]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(t * N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3400 or 500 < 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 t around 0 13.9%

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

      1. associate-*r/ 13.9%

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

      2. neg-mul-1 13.9%

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

      3. distribute-lft-neg-in 13.9%

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

   5. Simplified 13.9%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 13.4%

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

      3. sqr-neg 13.4%

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

      4. sqrt-unprod 7.9%

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

      5. add-sqr-sqrt 13.9%

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

      6. pow1 13.9%

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

   7. Applied egg-rr 13.9%

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

      1. unpow1 13.9%

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

   9. Simplified 13.9%

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

   if -3400 < t < 500

   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-sqr-sqrt 46.3%

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

      2. pow2 46.3%

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

   4. Applied egg-rr 52.5%

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

   5. Taylor expanded in t around 0 59.8%

      \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew + -0.5 \cdot \frac{{eh}^{2}}{ew}\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. distribute-lft-out 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \color{blue}{\left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right)} - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right| \]

      2. associate-*r/ 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}\right)\right| \]

      3. mul-1-neg 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right)\right)\right| \]

      4. distribute-lft-neg-out 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot \tan t}}{ew}\right)\right)\right| \]

   7. Simplified 59.8%

      \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   8. Taylor expanded in ew around inf 97.7%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \color{blue}{\left(-0.5 \cdot ew\right)} - eh \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right| \]

   9. Taylor expanded in t around 0 97.7%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right)\right| \]
   10. Step-by-step derivation

       1. associate-*r/ 97.7%

          \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

       2. associate-*r* 97.7%

          \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

       3. mul-1-neg 97.7%

          \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

   11. Simplified 97.7%

       \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot t}{ew}\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3400 \lor \neg \left(t \leq 500\right):\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew + (eh * (sin(t) * sin(atan(((t * eh) / 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 + (eh * (sin(t) * sin(atan(((t * eh) / ew)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $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. 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. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. add-sqr-sqrt 47.6%

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

   7. sqrt-unprod 91.0%

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

   8. sqr-neg 91.0%

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

   9. sqrt-unprod 52.2%

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

   10. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 98.9%

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

6. Taylor expanded in t around 0 76.6%

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

7. Final simplification 76.6%

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

Alternative 7: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -31 \lor \neg \left(t \leq 6200\right):\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -31.0) (not (<= t 6200.0)))
   (fabs (* ew (/ 1.0 (hypot 1.0 (* (tan t) (/ eh ew))))))
   (fabs
    (+
     ew
     (* t (- (* t (* ew -0.5)) (* eh (sin (atan (/ (* t eh) (- ew)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -31.0) || !(t <= 6200.0)) {
		tmp = fabs((ew * (1.0 / hypot(1.0, (tan(t) * (eh / ew))))));
	} else {
		tmp = fabs((ew + (t * ((t * (ew * -0.5)) - (eh * sin(atan(((t * eh) / -ew))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -31.0) || !(t <= 6200.0)) {
		tmp = Math.abs((ew * (1.0 / Math.hypot(1.0, (Math.tan(t) * (eh / ew))))));
	} else {
		tmp = Math.abs((ew + (t * ((t * (ew * -0.5)) - (eh * Math.sin(Math.atan(((t * eh) / -ew))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -31.0) or not (t <= 6200.0):
		tmp = math.fabs((ew * (1.0 / math.hypot(1.0, (math.tan(t) * (eh / ew))))))
	else:
		tmp = math.fabs((ew + (t * ((t * (ew * -0.5)) - (eh * math.sin(math.atan(((t * eh) / -ew))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -31.0) || !(t <= 6200.0))
		tmp = abs(Float64(ew * Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew))))));
	else
		tmp = abs(Float64(ew + Float64(t * Float64(Float64(t * Float64(ew * -0.5)) - Float64(eh * sin(atan(Float64(Float64(t * eh) / Float64(-ew)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -31.0) || ~((t <= 6200.0)))
		tmp = abs((ew * (1.0 / hypot(1.0, (tan(t) * (eh / ew))))));
	else
		tmp = abs((ew + (t * ((t * (ew * -0.5)) - (eh * sin(atan(((t * eh) / -ew))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -31.0], N[Not[LessEqual[t, 6200.0]], $MachinePrecision]], N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(t * N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -31 or 6200 < 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 t around 0 13.9%

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

      1. associate-*r/ 13.9%

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

      2. neg-mul-1 13.9%

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

      3. distribute-lft-neg-in 13.9%

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

   5. Simplified 13.9%

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

      1. cos-atan 13.5%

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

      2. hypot-1-def 13.5%

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

      3. associate-/l* 13.5%

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

      4. add-sqr-sqrt 5.9%

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

      5. sqrt-unprod 12.5%

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

      6. sqr-neg 12.5%

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

      7. sqrt-unprod 7.7%

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

      8. add-sqr-sqrt 13.5%

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

   7. Applied egg-rr 13.5%

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

      1. *-commutative 13.5%

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

      2. associate-*l/ 13.5%

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

      3. associate-*r/ 13.5%

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

   9. Simplified 13.5%

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

   if -31 < t < 6200

   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-sqr-sqrt 46.3%

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

      2. pow2 46.3%

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

   4. Applied egg-rr 52.5%

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

   5. Taylor expanded in t around 0 59.8%

      \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew + -0.5 \cdot \frac{{eh}^{2}}{ew}\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. distribute-lft-out 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \color{blue}{\left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right)} - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right| \]

      2. associate-*r/ 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}\right)\right| \]

      3. mul-1-neg 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right)\right)\right| \]

      4. distribute-lft-neg-out 59.8%

         \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot \tan t}}{ew}\right)\right)\right| \]

   7. Simplified 59.8%

      \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   8. Taylor expanded in ew around inf 97.7%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \color{blue}{\left(-0.5 \cdot ew\right)} - eh \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right| \]

   9. Taylor expanded in t around 0 97.7%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right)\right| \]
   10. Step-by-step derivation

       1. associate-*r/ 97.7%

          \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

       2. associate-*r* 97.7%

          \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

       3. mul-1-neg 97.7%

          \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

   11. Simplified 97.7%

       \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot t}{ew}\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -31 \lor \neg \left(t \leq 6200\right):\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.2% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* ew (/ 1.0 (hypot 1.0 (* (tan t) (/ eh ew)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 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. Taylor expanded in t around 0 41.9%

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

   1. associate-*r/ 41.9%

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

   2. neg-mul-1 41.9%

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

   3. distribute-lft-neg-in 41.9%

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

5. Simplified 41.9%

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

   1. cos-atan 41.5%

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

   2. hypot-1-def 41.6%

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

   3. associate-/l* 41.6%

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

   4. add-sqr-sqrt 19.5%

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

   5. sqrt-unprod 35.5%

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

   6. sqr-neg 35.5%

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

   7. sqrt-unprod 22.1%

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

   8. add-sqr-sqrt 41.6%

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

7. Applied egg-rr 41.6%

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

   1. *-commutative 41.6%

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

   2. associate-*l/ 41.6%

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

   3. associate-*r/ 41.6%

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

9. Simplified 41.6%

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

Alternative 9: 41.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return cos(atan((t * (eh / ew)))) * 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 = cos(atan((t * (eh / ew)))) * abs(ew)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(N[Cos[N[ArcTan[N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Abs[ew], $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. Taylor expanded in t around 0 41.9%

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

   1. associate-*r/ 41.9%

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

   2. neg-mul-1 41.9%

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

   3. distribute-lft-neg-in 41.9%

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

5. Simplified 41.9%

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

6. Taylor expanded in t around 0 40.6%

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

   1. associate-*r/ 52.1%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

   2. associate-*r* 52.1%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

   3. mul-1-neg 52.1%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

8. Simplified 40.6%

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

   1. add-sqr-sqrt 40.4%

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

   2. pow2 40.4%

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

   3. associate-/l* 40.4%

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

10. Applied egg-rr 40.4%

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

    1. *-un-lft-identity 40.4%

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

    2. unpow2 40.4%

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

    3. add-sqr-sqrt 40.6%

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

    4. distribute-lft-neg-out 40.6%

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

    5. associate-/l* 40.6%

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

    6. atan-neg 40.6%

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

    7. associate-/l* 40.6%

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

12. Applied egg-rr 40.6%

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

    1. *-lft-identity 40.6%

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

    2. fabs-mul 40.6%

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

    3. *-commutative 40.6%

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

    4. rem-square-sqrt 40.6%

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

    5. fabs-sqr 40.6%

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

    6. rem-square-sqrt 40.6%

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

    7. cos-neg 40.6%

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

    8. associate-*r/ 40.6%

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

    9. *-commutative 40.6%

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

    10. associate-/l* 40.6%

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

14. Simplified 40.6%

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

Alternative 10: 40.6% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* ew (/ 1.0 (hypot 1.0 (* eh (/ t (- ew))))))))

C

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

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * (1.0 / Math.hypot(1.0, (eh * (t / -ew))))));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * (1.0 / math.hypot(1.0, (eh * (t / -ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * Float64(1.0 / hypot(1.0, Float64(eh * Float64(t / Float64(-ew)))))))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh * N[(t / (-ew)), $MachinePrecision]), $MachinePrecision] ^ 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. Taylor expanded in t around 0 41.9%

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

   1. associate-*r/ 41.9%

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

   2. neg-mul-1 41.9%

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

   3. distribute-lft-neg-in 41.9%

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

5. Simplified 41.9%

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

6. Taylor expanded in t around 0 40.6%

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

   1. associate-*r/ 52.1%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

   2. associate-*r* 52.1%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

   3. mul-1-neg 52.1%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

8. Simplified 40.6%

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

   1. cos-atan 39.6%

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

   2. hypot-1-def 39.6%

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

   3. associate-/l* 39.7%

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

10. Applied egg-rr 39.7%

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

    1. *-commutative 39.7%

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

12. Simplified 39.7%

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

13. Final simplification 39.7%

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

Alternative 11: 39.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \left|ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (+ ew (* -0.5 (* ew (* t t))))))

C

double code(double eh, double ew, double t) {
	return fabs((ew + (-0.5 * (ew * (t * 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 + ((-0.5d0) * (ew * (t * t)))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew + (-0.5 * (ew * (t * t)))));
}

Python

def code(eh, ew, t):
	return math.fabs((ew + (-0.5 * (ew * (t * t)))))

Julia

function code(eh, ew, t)
	return abs(Float64(ew + Float64(-0.5 * Float64(ew * Float64(t * t)))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew + (-0.5 * (ew * (t * t)))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew + N[(-0.5 * N[(ew * N[(t * t), $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. add-sqr-sqrt 47.0%

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

   2. pow2 47.0%

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

4. Applied egg-rr 56.4%

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

5. Taylor expanded in t around 0 32.8%

   \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(-0.5 \cdot ew + -0.5 \cdot \frac{{eh}^{2}}{ew}\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
6. Step-by-step derivation

   1. distribute-lft-out 32.8%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \color{blue}{\left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right)} - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right| \]

   2. associate-*r/ 32.8%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}\right)\right| \]

   3. mul-1-neg 32.8%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right)\right)\right| \]

   4. distribute-lft-neg-out 32.8%

      \[\leadsto \left|ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot \tan t}}{ew}\right)\right)\right| \]

7. Simplified 32.8%

   \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(-0.5 \cdot \left(ew + \frac{{eh}^{2}}{ew}\right)\right) - eh \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

8. Taylor expanded in ew around inf 37.8%

   \[\leadsto \left|ew + \color{blue}{-0.5 \cdot \left(ew \cdot {t}^{2}\right)}\right| \]
9. Step-by-step derivation

   1. *-commutative 37.8%

      \[\leadsto \left|ew + -0.5 \cdot \color{blue}{\left({t}^{2} \cdot ew\right)}\right| \]

10. Simplified 37.8%

    \[\leadsto \left|ew + \color{blue}{-0.5 \cdot \left({t}^{2} \cdot ew\right)}\right| \]
11. Step-by-step derivation

    1. unpow2 37.8%

       \[\leadsto \left|ew + -0.5 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot ew\right)\right| \]

12. Applied egg-rr 37.8%

    \[\leadsto \left|ew + -0.5 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot ew\right)\right| \]

13. Final simplification 37.8%

    \[\leadsto \left|ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right| \]
14. Add Preprocessing

Reproduce

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