Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.3s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

9. Taylor expanded in ew around 0 97.7%

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

10. Final simplification 97.7%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

9. Taylor expanded in eh around 0 97.7%

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

    1. *-commutative 97.7%

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

    2. mul-1-neg 97.7%

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

    3. associate-*l/ 97.7%

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

    4. *-commutative 97.7%

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

    5. distribute-rgt-neg-in 97.7%

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

11. Simplified 97.7%

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

12. Taylor expanded in ew around 0 97.7%

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

13. Final simplification 97.7%

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

Alternative 4: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

9. Taylor expanded in t around 0 97.3%

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

    1. mul-1-neg 97.3%

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

    2. associate-/l* 97.3%

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

    3. distribute-neg-frac 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 5: 98.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

   1. associate-*r* 97.7%

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

   2. sin-atan 76.3%

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

   3. associate-*r/ 75.3%

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

   4. div-inv 75.3%

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

   5. add-sqr-sqrt 38.2%

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

   6. sqrt-unprod 65.2%

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

   7. sqr-neg 65.2%

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

   8. sqrt-unprod 36.9%

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

   9. add-sqr-sqrt 75.2%

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

   10. clear-num 75.2%

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

   11. hypot-1-def 80.9%

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

   12. div-inv 80.9%

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

10. Applied egg-rr 81.0%

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

    1. associate-/l* 89.0%

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

    2. associate-/r/ 89.0%

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

    3. *-commutative 89.0%

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

    4. associate-*l/ 88.9%

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

    5. associate-*r/ 83.3%

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

    6. *-commutative 83.3%

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

    7. associate-*l/ 83.3%

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

    8. associate-*r/ 83.2%

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

12. Simplified 83.2%

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

13. Taylor expanded in eh around inf 97.1%

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

14. Final simplification 97.1%

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

Alternative 6: 98.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

   1. associate-*r* 97.7%

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

   2. sin-atan 76.3%

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

   3. associate-*r/ 75.3%

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

   4. div-inv 75.3%

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

   5. add-sqr-sqrt 38.2%

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

   6. sqrt-unprod 65.2%

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

   7. sqr-neg 65.2%

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

   8. sqrt-unprod 36.9%

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

   9. add-sqr-sqrt 75.2%

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

   10. clear-num 75.2%

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

   11. hypot-1-def 80.9%

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

   12. div-inv 80.9%

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

10. Applied egg-rr 81.0%

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

    1. associate-/l* 89.0%

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

    2. associate-/r/ 89.0%

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

    3. *-commutative 89.0%

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

    4. associate-*l/ 88.9%

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

    5. associate-*r/ 83.3%

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

    6. *-commutative 83.3%

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

    7. associate-*l/ 83.3%

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

    8. associate-*r/ 83.2%

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

12. Simplified 83.2%

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

13. Taylor expanded in eh around -inf 97.1%

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

    1. mul-1-neg 97.1%

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

    2. distribute-rgt-neg-in 97.1%

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

15. Simplified 97.1%

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

16. Final simplification 97.1%

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

Alternative 7: 61.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

10. Applied egg-rr 57.6%

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

    1. +-inverses 57.6%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

    2. *-commutative 57.6%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

    3. associate-/l* 57.6%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

    4. div0 57.6%

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

12. Simplified 57.6%

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

13. Taylor expanded in ew around 0 57.6%

    \[\leadsto \left|\color{blue}{\cos t \cdot ew} - 0\right| \]

14. Final simplification 57.6%

    \[\leadsto \left|ew \cdot \cos t\right| \]

Alternative 8: 42.0% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--r+ 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 50.3%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 97.7%

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

10. Applied egg-rr 57.6%

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

    1. +-inverses 57.6%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

    2. *-commutative 57.6%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

    3. associate-/l* 57.6%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

    4. div0 57.6%

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

12. Simplified 57.6%

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

13. Taylor expanded in t around 0 37.0%

    \[\leadsto \left|\color{blue}{ew} - 0\right| \]

14. Final simplification 37.0%

    \[\leadsto \left|ew\right| \]

Reproduce

herbie shell --seed 2023200 
(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)))))))