Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.7s

Alternatives: 5

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

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

FPCore

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

C

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

Java

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

Python

def code(eh, ew, t):
	return math.fabs(((ew * (math.cos(t) * (1.0 / math.hypot(1.0, (math.tan(t) * (eh / ew)))))) - (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(cos(t) * Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))))) - Float64(eh * Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. fabs-neg 99.8%

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

   3. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   4. associate-+l- 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   5. neg-sub0 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 53.4%

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

   4. sqrt-unprod 93.4%

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

   5. sqr-neg 93.4%

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

   6. sqrt-unprod 46.4%

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

   7. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-/l* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. fabs-neg 99.8%

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

   3. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   4. associate-+l- 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   5. neg-sub0 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 99.5%

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

   1. associate-*r/ 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. fabs-sub 99.8%

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

   2. fabs-neg 99.8%

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

   3. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   4. associate-+l- 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   5. neg-sub0 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 81.2%

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

   3. associate-*r/ 78.5%

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

   4. add-sqr-sqrt 41.9%

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

   5. sqrt-unprod 65.5%

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

   6. sqr-neg 65.5%

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

   7. sqrt-unprod 36.3%

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

   8. add-sqr-sqrt 78.0%

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

   9. hypot-1-def 85.0%

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

   10. add-sqr-sqrt 44.3%

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

   11. sqrt-unprod 74.1%

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

   12. sqr-neg 74.1%

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

   13. sqrt-unprod 40.5%

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

6. Applied egg-rr 85.0%

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

7. Taylor expanded in eh around inf 99.4%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 53.4%

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

   4. sqrt-unprod 93.4%

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

   5. sqr-neg 93.4%

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

   6. sqrt-unprod 46.4%

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

   7. add-sqr-sqrt 99.8%

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

9. Applied egg-rr 99.4%

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

    1. *-commutative 99.8%

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

    2. associate-*l/ 99.8%

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

    3. associate-/l* 99.8%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 98.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - (eh * sin(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)) - (eh * sin(t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. fabs-sub 99.8%

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

   2. fabs-neg 99.8%

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

   3. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   4. associate-+l- 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   5. neg-sub0 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 81.2%

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

   3. associate-*r/ 78.5%

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

   4. add-sqr-sqrt 41.9%

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

   5. sqrt-unprod 65.5%

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

   6. sqr-neg 65.5%

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

   7. sqrt-unprod 36.3%

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

   8. add-sqr-sqrt 78.0%

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

   9. hypot-1-def 85.0%

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

   10. add-sqr-sqrt 44.3%

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

   11. sqrt-unprod 74.1%

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

   12. sqr-neg 74.1%

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

   13. sqrt-unprod 40.5%

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

6. Applied egg-rr 85.0%

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

7. Taylor expanded in eh around inf 99.4%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 53.4%

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

   4. sqrt-unprod 93.4%

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

   5. sqr-neg 93.4%

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

   6. sqrt-unprod 46.4%

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

   7. add-sqr-sqrt 99.8%

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

9. Applied egg-rr 99.4%

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

    1. *-commutative 99.8%

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

    2. associate-*l/ 99.8%

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

    3. associate-/l* 99.8%

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

11. Simplified 99.4%

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

12. Taylor expanded in ew around inf 99.1%

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

13. Final simplification 99.1%

    \[\leadsto \left|ew \cdot \cos t - eh \cdot \sin t\right| \]
14. Add Preprocessing

Alternative 5: 79.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|ew - eh \cdot \sin t\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew - (eh * sin(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 - (eh * sin(t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - N[(eh * N[Sin[t], $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. Step-by-step derivation

   1. fabs-sub 99.8%

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

   2. fabs-neg 99.8%

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

   3. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   4. associate-+l- 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   5. neg-sub0 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 81.2%

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

   3. associate-*r/ 78.5%

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

   4. add-sqr-sqrt 41.9%

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

   5. sqrt-unprod 65.5%

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

   6. sqr-neg 65.5%

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

   7. sqrt-unprod 36.3%

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

   8. add-sqr-sqrt 78.0%

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

   9. hypot-1-def 85.0%

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

   10. add-sqr-sqrt 44.3%

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

   11. sqrt-unprod 74.1%

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

   12. sqr-neg 74.1%

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

   13. sqrt-unprod 40.5%

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

6. Applied egg-rr 85.0%

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

7. Taylor expanded in eh around inf 99.4%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 53.4%

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

   4. sqrt-unprod 93.4%

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

   5. sqr-neg 93.4%

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

   6. sqrt-unprod 46.4%

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

   7. add-sqr-sqrt 99.8%

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

9. Applied egg-rr 99.4%

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

    1. *-commutative 99.8%

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

    2. associate-*l/ 99.8%

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

    3. associate-/l* 99.8%

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

11. Simplified 99.4%

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

12. Taylor expanded in t around 0 79.1%

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

13. Final simplification 79.1%

    \[\leadsto \left|ew - eh \cdot \sin t\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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)))))))