Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.5s

Alternatives: 6

Speedup: 1.1×

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan t \cdot eh\\ \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \frac{t\_1}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{0 - ew}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tan t) eh)))
   (fabs
    (-
     (/ (* ew (cos t)) (hypot 1.0 (/ t_1 ew)))
     (* (* eh (sin t)) (sin (atan (/ t_1 (- 0.0 ew)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) * eh;
	return fabs((((ew * cos(t)) / hypot(1.0, (t_1 / ew))) - ((eh * sin(t)) * sin(atan((t_1 / (0.0 - ew)))))));
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.tan(t) * eh;
	return Math.abs((((ew * Math.cos(t)) / Math.hypot(1.0, (t_1 / ew))) - ((eh * Math.sin(t)) * Math.sin(Math.atan((t_1 / (0.0 - ew)))))));
}

Python

def code(eh, ew, t):
	t_1 = math.tan(t) * eh
	return math.fabs((((ew * math.cos(t)) / math.hypot(1.0, (t_1 / ew))) - ((eh * math.sin(t)) * math.sin(math.atan((t_1 / (0.0 - ew)))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) * eh)
	return abs(Float64(Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(t_1 / ew))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(t_1 / Float64(0.0 - ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = tan(t) * eh;
	tmp = abs((((ew * cos(t)) / hypot(1.0, (t_1 / ew))) - ((eh * sin(t)) * sin(atan((t_1 / (0.0 - ew)))))));
end

Wolfram

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

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

   5. unpow-prod-down N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. unpow-1 N/A

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

   3. un-div-inv N/A

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

   4. unpow-1 N/A

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

   5. times-frac N/A

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

   6. pow-prod-up N/A

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

   7. metadata-eval N/A

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

   8. pow2 N/A

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

   9. metadata-eval N/A

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

   10. pow1/2 N/A

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(0.0 - 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. 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 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 98.5% 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. 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. sqr-pow N/A

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

   5. unpow-prod-down N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Taylor expanded in ew around inf

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

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

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

   2. cos-lowering-cos.f64 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 4: 85.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(0 - \frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{if}\;eh \leq -920000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.9 \cdot 10^{+46}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (- ew (* (* eh (sin t)) (sin (atan (- 0.0 (/ (* t eh) ew)))))))))
   (if (<= eh -920000000.0)
     t_1
     (if (<= eh 3.9e+46) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew - ((eh * sin(t)) * sin(atan((0.0 - ((t * eh) / ew)))))));
	double tmp;
	if (eh <= -920000000.0) {
		tmp = t_1;
	} else if (eh <= 3.9e+46) {
		tmp = fabs((ew * cos(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 - ((eh * sin(t)) * sin(atan((0.0d0 - ((t * eh) / ew)))))))
    if (eh <= (-920000000.0d0)) then
        tmp = t_1
    else if (eh <= 3.9d+46) then
        tmp = abs((ew * cos(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 - ((eh * Math.sin(t)) * Math.sin(Math.atan((0.0 - ((t * eh) / ew)))))));
	double tmp;
	if (eh <= -920000000.0) {
		tmp = t_1;
	} else if (eh <= 3.9e+46) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew - ((eh * math.sin(t)) * math.sin(math.atan((0.0 - ((t * eh) / ew)))))))
	tmp = 0
	if eh <= -920000000.0:
		tmp = t_1
	elif eh <= 3.9e+46:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew - Float64(Float64(eh * sin(t)) * sin(atan(Float64(0.0 - Float64(Float64(t * eh) / ew)))))))
	tmp = 0.0
	if (eh <= -920000000.0)
		tmp = t_1;
	elseif (eh <= 3.9e+46)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew - ((eh * sin(t)) * sin(atan((0.0 - ((t * eh) / ew)))))));
	tmp = 0.0;
	if (eh <= -920000000.0)
		tmp = t_1;
	elseif (eh <= 3.9e+46)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(0.0 - N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -920000000.0], t$95$1, If[LessEqual[eh, 3.9e+46], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.2e8 or 3.89999999999999995e46 < eh

   1. Initial program 99.9%

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

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

      5. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

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

   7. Simplified 93.3%

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

   8. Taylor expanded in t around 0

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

   10. Simplified 93.3%

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

   if -9.2e8 < eh < 3.89999999999999995e46

   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. sqr-pow N/A

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

      5. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   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. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -920000000:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(0 - \frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 3.9 \cdot 10^{+46}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(0 - \frac{t \cdot eh}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. sqr-pow N/A

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

   5. unpow-prod-down N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

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. *-lowering-*.f64 N/A

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

   2. cos-lowering-cos.f64 62.7%

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

7. Simplified 62.7%

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

Alternative 6: 41.9% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
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. sqr-pow N/A

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

   5. unpow-prod-down N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Taylor expanded in t around 0

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

7. Simplified 43.3%

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

Reproduce

herbie shell --seed 2024156 
(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)))))))