Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.5s

Alternatives: 9

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, 1.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (tan t) (/ ew (- eh))))))
   (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((tan(t) / (ew / -eh)));
	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((tan(t) / (ew / -eh)))
    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((Math.tan(t) / (ew / -eh)));
	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((math.tan(t) / (ew / -eh)))
	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(tan(t) / Float64(ew / Float64(-eh))))
	return abs(Float64(Float64(ew * Float64(cos(t) * cos(t_1))) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan((tan(t) / (ew / -eh)));
	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[Tan[t], $MachinePrecision] / N[(ew / (-eh)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $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. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

   1. associate-/r/ 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l/ 99.8%

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

   4. associate-*r/ 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 58.1%

      \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right) \cdot \left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right) \cdot \left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. pow3 58.0%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)}^{3}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 59.1%

   \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}^{3}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. rem-cbrt-cube 99.8%

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

   2. associate-/l* 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in t around 0 99.2%

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

   1. mul-1-neg 77.1%

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

   2. associate-/l* 77.1%

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

   3. distribute-neg-frac 77.1%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 4: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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) / (ew / -eh)))))))
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) / (ew / -eh)))))));
}

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan((tan(t) / (ew / -eh)))))));
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[Tan[t], $MachinePrecision] / N[(ew / (-eh)), $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. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in eh around 0 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $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. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

   1. associate-/r/ 99.7%

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

   2. add-sqr-sqrt 49.8%

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

   3. sqrt-unprod 95.8%

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

   4. sqr-neg 95.8%

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

   5. sqrt-unprod 49.5%

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

   6. add-sqr-sqrt 99.1%

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in eh around 0 99.1%

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

11. Final simplification 99.1%

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

Alternative 6: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(ew * N[Cos[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. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

   1. associate-/r/ 99.7%

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

   2. add-sqr-sqrt 49.8%

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

   3. sqrt-unprod 95.8%

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

   4. sqr-neg 95.8%

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

   5. sqrt-unprod 49.5%

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

   6. add-sqr-sqrt 99.1%

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in t around 0 98.9%

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

11. Taylor expanded in eh around 0 98.8%

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

12. Final simplification 98.8%

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

Alternative 7: 78.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 77.1%

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

9. Taylor expanded in t around 0 77.1%

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

    1. mul-1-neg 77.1%

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

    2. associate-/l* 77.1%

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

    3. distribute-neg-frac 77.1%

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

11. Simplified 77.1%

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

12. Final simplification 77.1%

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

Alternative 8: 78.7% 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. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 77.1%

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

   1. sin-atan 56.5%

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

   2. associate-*r/ 54.8%

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

   3. associate-/r/ 54.9%

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

   4. add-sqr-sqrt 26.2%

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

   5. sqrt-unprod 45.9%

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

   6. sqr-neg 45.9%

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

   7. sqrt-unprod 28.6%

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

   8. add-sqr-sqrt 54.9%

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

   9. associate-*l/ 54.8%

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

   10. associate-*r/ 54.7%

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

   11. hypot-1-def 58.4%

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

   12. associate-/r/ 58.6%

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

   13. add-sqr-sqrt 27.9%

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

   14. sqrt-unprod 51.0%

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

   15. sqr-neg 51.0%

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

   16. sqrt-unprod 30.6%

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

   17. add-sqr-sqrt 58.6%

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

10. Applied egg-rr 58.5%

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

    1. *-commutative 58.5%

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

    2. associate-*r* 57.3%

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

    3. associate-*r/ 57.4%

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

    4. associate-*l/ 57.3%

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

    5. associate-*l* 49.2%

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

    6. unpow2 49.2%

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

    7. *-commutative 49.2%

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

    8. associate-/r/ 50.0%

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

12. Simplified 50.0%

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

13. Taylor expanded in ew around 0 77.0%

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

14. Final simplification 77.0%

    \[\leadsto \left|ew - eh \cdot \sin t\right| \]

Alternative 9: 78.7% 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. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 70.1%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-udef 53.9%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

5. Applied egg-rr 57.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 73.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right| \]

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/l* 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*l/ 99.7%

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

   7. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 77.1%

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

   1. sin-atan 56.5%

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

   2. associate-*r/ 54.8%

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

   3. associate-/r/ 54.9%

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

   4. add-sqr-sqrt 26.2%

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

   5. sqrt-unprod 45.9%

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

   6. sqr-neg 45.9%

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

   7. sqrt-unprod 28.6%

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

   8. add-sqr-sqrt 54.9%

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

   9. associate-*l/ 54.8%

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

   10. associate-*r/ 54.7%

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

   11. hypot-1-def 58.4%

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

   12. associate-/r/ 58.6%

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

   13. add-sqr-sqrt 27.9%

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

   14. sqrt-unprod 51.0%

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

   15. sqr-neg 51.0%

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

   16. sqrt-unprod 30.6%

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

   17. add-sqr-sqrt 58.6%

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

10. Applied egg-rr 58.5%

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

    1. *-commutative 58.5%

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

    2. associate-*r* 57.3%

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

    3. associate-*r/ 57.4%

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

    4. associate-*l/ 57.3%

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

    5. associate-*l* 49.2%

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

    6. unpow2 49.2%

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

    7. *-commutative 49.2%

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

    8. associate-/r/ 50.0%

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

12. Simplified 50.0%

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

13. Taylor expanded in eh around -inf 77.0%

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

    1. mul-1-neg 77.0%

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

    2. *-commutative 77.0%

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

    3. distribute-rgt-neg-in 77.0%

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

15. Simplified 77.0%

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

16. Final simplification 77.0%

    \[\leadsto \left|ew + eh \cdot \sin t\right| \]

Reproduce

herbie shell --seed 2023307 
(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)))))))