Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.0s

Alternatives: 8

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * (cos(t) * (1.0 / hypot(1.0, ((tan(t) / ew) * eh))))) - (sin(t) * (eh * sin(atan((tan(t) * (eh / -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[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-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. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.4%

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

   1. mul-1-neg 74.8%

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

   2. associate-/l* 74.8%

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

   3. distribute-neg-frac 74.8%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.4%

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

   1. mul-1-neg 74.8%

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

   2. associate-/l* 74.8%

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

   3. distribute-neg-frac 74.8%

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

6. Simplified 99.4%

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

   1. associate-/r/ 99.4%

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

   2. add-sqr-sqrt 49.1%

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

   3. sqrt-unprod 97.7%

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

   4. sqr-neg 97.7%

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

   5. sqrt-unprod 50.0%

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

   6. add-sqr-sqrt 99.2%

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

8. Applied egg-rr 99.2%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

10. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.8%

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

    2. associate-*l/ 99.8%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.9%

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

9. Taylor expanded in ew around 0 98.9%

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

10. Final simplification 98.9%

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

Alternative 5: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.4%

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

   1. mul-1-neg 74.8%

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

   2. associate-/l* 74.8%

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

   3. distribute-neg-frac 74.8%

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

6. Simplified 99.4%

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

   1. associate-/r/ 99.4%

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

   2. add-sqr-sqrt 49.1%

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

   3. sqrt-unprod 97.7%

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

   4. sqr-neg 97.7%

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

   5. sqrt-unprod 50.0%

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

   6. add-sqr-sqrt 99.2%

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

8. Applied egg-rr 99.2%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

10. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.8%

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

    2. associate-*l/ 99.8%

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

12. Simplified 99.2%

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

13. Taylor expanded in t around 0 98.8%

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

14. Final simplification 98.8%

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

Alternative 6: 79.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.9%

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

9. Taylor expanded in t around 0 74.8%

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

    1. add-log-exp 40.0%

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

    2. *-un-lft-identity 40.0%

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

    3. log-prod 40.0%

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

    4. metadata-eval 40.0%

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

    5. add-log-exp 74.8%

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

    6. *-commutative 74.8%

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

    7. associate-*l* 74.8%

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

11. Applied egg-rr 74.8%

    \[\leadsto \left|ew - \color{blue}{\left(0 + eh \cdot \left(\sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right) \cdot \sin t\right)\right)}\right| \]
12. Step-by-step derivation

    1. +-lft-identity 74.8%

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

    2. *-commutative 74.8%

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

    3. associate-*r* 74.8%

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

    4. associate-*r/ 74.8%

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

    5. associate-*l/ 74.8%

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

    6. *-commutative 74.8%

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

13. Simplified 74.8%

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

14. Final simplification 74.8%

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

Alternative 7: 54.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.9%

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

9. Taylor expanded in t around 0 74.8%

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

10. Taylor expanded in t around 0 54.3%

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

    1. mul-1-neg 54.3%

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

    2. *-commutative 54.3%

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

    3. associate-*l/ 54.3%

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

    4. distribute-rgt-neg-in 54.3%

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

12. Simplified 54.3%

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

13. Final simplification 54.3%

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

Alternative 8: 79.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 51.1%

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

   6. sqrt-unprod 94.4%

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

   7. sqr-neg 94.4%

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

   8. sqrt-unprod 48.8%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.9%

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

9. Taylor expanded in t around 0 74.8%

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

10. Taylor expanded in t around 0 74.8%

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

    1. mul-1-neg 74.8%

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

    2. associate-/l* 74.8%

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

    3. distribute-neg-frac 74.8%

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

12. Simplified 74.8%

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

13. Final simplification 74.8%

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

Reproduce

herbie shell --seed 2023338 
(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)))))))