Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.5s

Alternatives: 7

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

\[\begin{array}{l} \\ \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{0 - ew}\right) - \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) eh) (- 0.0 ew)))))
   (/ (* ew (cos t)) (hypot 1.0 (* (tan t) (/ eh ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((eh * sin(t)) * sin(atan(((tan(t) * eh) / (0.0 - ew))))) - ((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((eh * Math.sin(t)) * Math.sin(Math.atan(((Math.tan(t) * eh) / (0.0 - ew))))) - ((ew * Math.cos(t)) / Math.hypot(1.0, (Math.tan(t) * (eh / ew))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((eh * math.sin(t)) * math.sin(math.atan(((math.tan(t) * eh) / (0.0 - ew))))) - ((ew * math.cos(t)) / math.hypot(1.0, (math.tan(t) * (eh / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * eh) / Float64(0.0 - ew))))) - Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(tan(t) * Float64(eh / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((tan(t) * eh) / (0.0 - ew))))) - ((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. flip-+ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\frac{1 \cdot 1 - \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}{1 - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\left(1 \cdot 1 - \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right) \cdot \frac{1}{1 - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(1 \cdot 1 - \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {\left(\frac{1}{1 - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 64.7%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left(1 - 1 \cdot {\left(\frac{\tan t \cdot eh}{ew}\right)}^{4}\right)}^{-0.5} \cdot {\left(\frac{1}{1 - {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}}\right)}^{-0.5}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \left({\left(\frac{1}{1 - {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}}\right)}^{\frac{-1}{2}} \cdot {\left(1 - 1 \cdot {\left(\frac{\tan t \cdot eh}{ew}\right)}^{4}\right)}^{\frac{-1}{2}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. pow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot {\left(\frac{1}{1 - {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}} \cdot \left(1 - 1 \cdot {\left(\frac{\tan t \cdot eh}{ew}\right)}^{4}\right)\right)}^{\frac{-1}{2}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{0 - ew}\right) - \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right| \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right) - \cos \tan^{-1} \left(\frac{\tan t \cdot eh}{0 - ew}\right) \cdot \left(ew \cdot \cos t\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* eh (sin t)) (sin (atan (/ (* t (- 0.0 eh)) ew))))
   (* (cos (atan (/ (* (tan t) eh) (- 0.0 ew)))) (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	return fabs((((eh * sin(t)) * sin(atan(((t * (0.0 - eh)) / ew)))) - (cos(atan(((tan(t) * eh) / (0.0 - ew)))) * (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((((eh * sin(t)) * sin(atan(((t * (0.0d0 - eh)) / ew)))) - (cos(atan(((tan(t) * eh) / (0.0d0 - ew)))) * (ew * cos(t)))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((eh * Math.sin(t)) * Math.sin(Math.atan(((t * (0.0 - eh)) / ew)))) - (Math.cos(Math.atan(((Math.tan(t) * eh) / (0.0 - ew)))) * (ew * Math.cos(t)))));
}

Python

def code(eh, ew, t):
	return math.fabs((((eh * math.sin(t)) * math.sin(math.atan(((t * (0.0 - eh)) / ew)))) - (math.cos(math.atan(((math.tan(t) * eh) / (0.0 - ew)))) * (ew * math.cos(t)))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(0.0 - eh)) / ew)))) - Float64(cos(atan(Float64(Float64(tan(t) * eh) / Float64(0.0 - ew)))) * Float64(ew * cos(t)))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((t * (0.0 - eh)) / ew)))) - (cos(atan(((tan(t) * eh) / (0.0 - ew)))) * (ew * cos(t)))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * N[(0.0 - eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(ew * N[Cos[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. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \color{blue}{t}\right), ew\right)\right)\right)\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 99.5%

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

6. Final simplification 99.5%

   \[\leadsto \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right) - \cos \tan^{-1} \left(\frac{\tan t \cdot eh}{0 - ew}\right) \cdot \left(ew \cdot \cos t\right)\right| \]
7. Add Preprocessing

Alternative 3: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{0 - ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* ew (cos t))
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) eh) (- 0.0 ew))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((tan(t) * eh) / (0.0 - 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(((tan(t) * eh) / (0.0d0 - 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(((Math.tan(t) * eh) / (0.0 - ew)))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((ew * math.cos(t)) - ((eh * math.sin(t)) * math.sin(math.atan(((math.tan(t) * eh) / (0.0 - ew)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * eh) / Float64(0.0 - ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((tan(t) * eh) / (0.0 - 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[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / N[(0.0 - 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. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. frac-2neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. sqr-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. hypot-1-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\mathsf{hypot}\left(1, \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) + ew \cdot \cos t\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) + ew \cdot \cos t\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(ew \cdot \cos t + \left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(ew \cdot \cos t\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\cos t \cdot ew\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos t, ew\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   7. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\left(eh \cdot \sin t\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\left(\sin t \cdot eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin t, eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   13. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   14. atan-lowering-atan.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

   15. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right)\right) \]

   17. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \left|ew \cdot \cos t - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{0 - ew}\right)\right| \]
9. Add Preprocessing

Alternative 4: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \color{blue}{t}\right), ew\right)\right)\right)\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 99.5%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \left(\left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right) \cdot eh\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right), eh\right)\right)\right) \]

7. Applied egg-rr 81.8%

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

9. Applied egg-rr 82.7%

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

10. Taylor expanded in eh around inf

    \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{tan.f64}\left(t\right), eh\right), ew\right)\right)\right), \mathsf{/.f64}\left(eh, \mathsf{neg.f64}\left(\color{blue}{\left(\frac{1}{\sin t}\right)}\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{tan.f64}\left(t\right), eh\right), ew\right)\right)\right), \mathsf{/.f64}\left(eh, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \sin t\right)\right)\right)\right)\right) \]

    2. sin-lowering-sin.f64 98.6%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{tan.f64}\left(t\right), eh\right), ew\right)\right)\right), \mathsf{/.f64}\left(eh, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right)\right) \]

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 5: 74.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -5 \cdot 10^{-124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.6 \cdot 10^{-97}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -5e-124) t_1 (if (<= ew 4.6e-97) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -5e-124) {
		tmp = t_1;
	} else if (ew <= 4.6e-97) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = 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 = abs((ew * cos(t)))
    if (ew <= (-5d-124)) then
        tmp = t_1
    else if (ew <= 4.6d-97) then
        tmp = abs((eh * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -5e-124) {
		tmp = t_1;
	} else if (ew <= 4.6e-97) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -5e-124:
		tmp = t_1
	elif ew <= 4.6e-97:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -5e-124)
		tmp = t_1;
	elseif (ew <= 4.6e-97)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -5e-124)
		tmp = t_1;
	elseif (ew <= 4.6e-97)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5e-124], t$95$1, If[LessEqual[ew, 4.6e-97], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.0000000000000003e-124 or 4.59999999999999988e-97 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      7. sqr-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\mathsf{hypot}\left(1, \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \cos t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\cos t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\cos t, ew\right)\right) \]

      3. cos-lowering-cos.f64 81.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right)\right) \]

   7. Simplified 81.4%

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

   if -5.0000000000000003e-124 < ew < 4.59999999999999988e-97

   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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \color{blue}{t}\right), ew\right)\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \left(\left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right) \cdot eh\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right), eh\right)\right)\right) \]

   7. Applied egg-rr 56.3%

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

   8. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin t\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 79.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   10. Simplified 79.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 4.6 \cdot 10^{-97}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-25}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -3.7e-17) t_1 (if (<= t 7.2e-25) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -3.7e-17) {
		tmp = t_1;
	} else if (t <= 7.2e-25) {
		tmp = fabs(ew);
	} else {
		tmp = 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 = abs((eh * sin(t)))
    if (t <= (-3.7d-17)) then
        tmp = t_1
    else if (t <= 7.2d-25) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -3.7e-17) {
		tmp = t_1;
	} else if (t <= 7.2e-25) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if t <= -3.7e-17:
		tmp = t_1
	elif t <= 7.2e-25:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -3.7e-17)
		tmp = t_1;
	elseif (t <= 7.2e-25)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (t <= -3.7e-17)
		tmp = t_1;
	elseif (t <= 7.2e-25)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.7e-17], t$95$1, If[LessEqual[t, 7.2e-25], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.6999999999999997e-17 or 7.1999999999999998e-25 < t

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \color{blue}{t}\right), ew\right)\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.1%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \left(\left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right) \cdot eh\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot t}{ew}\right)\right), eh\right)\right)\right) \]

   7. Applied egg-rr 67.6%

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

   8. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin t\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 51.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   10. Simplified 51.1%

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

   if -3.6999999999999997e-17 < t < 7.1999999999999998e-25

   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. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      7. sqr-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\mathsf{hypot}\left(1, \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
   6. Step-by-step derivation

   7. Simplified 80.7%

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

Alternative 7: 41.4% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. frac-2neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{\mathsf{neg}\left(ew\right)}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. sqr-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{1 + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. hypot-1-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \left(\mathsf{hypot}\left(1, \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
6. Step-by-step derivation

7. Simplified 45.1%

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

Reproduce

herbie shell --seed 2024139 
(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)))))))