Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.2s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \frac{\tan t}{ew}\\ \left|\mathsf{fma}\left(\left(-eh\right) \cdot \sin \tan^{-1} t\_1, \sin t, \frac{ew \cdot \cos t}{\sqrt{{t\_1}^{2} + 1}}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (/ (tan t) ew))))
   (fabs
    (fma
     (* (- eh) (sin (atan t_1)))
     (sin t)
     (/ (* ew (cos t)) (sqrt (+ (pow t_1 2.0) 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * (tan(t) / ew);
	return fabs(fma((-eh * sin(atan(t_1))), sin(t), ((ew * cos(t)) / sqrt((pow(t_1, 2.0) + 1.0)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * Float64(tan(t) / ew))
	return abs(fma(Float64(Float64(-eh) * sin(atan(t_1))), sin(t), Float64(Float64(ew * cos(t)) / sqrt(Float64((t_1 ^ 2.0) + 1.0)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[((-eh) * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lift-neg.f64 N/A

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

   10. associate-*r* N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \left|\mathsf{fma}\left(\left(-eh\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right), \sin t, \frac{ew \cdot \cos t}{\sqrt{{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2} + 1}}\right)\right| \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((ew / (1.0d0 / cos(t))) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. div-inv N/A

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

   7. atan-neg N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. div-inv N/A

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

   12. lift-/.f64 N/A

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

   13. lower-atan.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024230 
(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)))))))