Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.8s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. hypot-1-def 99.8%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.5%

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

   8. sqr-neg 94.5%

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

   9. sqrt-unprod 50.3%

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

   10. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/l* 99.8%

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

   6. hypot-1-def 99.8%

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

   7. add-sqr-sqrt 49.5%

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

   8. sqrt-unprod 94.5%

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

   9. sqr-neg 94.5%

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

   10. sqrt-unprod 50.3%

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

   11. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. associate-/l* 99.6%

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

   5. hypot-1-def 99.6%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.4%

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

   8. sqr-neg 94.4%

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

   9. sqrt-unprod 50.2%

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

   10. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in t around 0 99.3%

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

   1. mul-1-neg 79.3%

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

   2. associate-/l* 79.3%

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

   3. associate-/r/ 79.3%

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

   4. distribute-rgt-neg-in 79.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. associate-/l* 99.6%

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

   5. hypot-1-def 99.6%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.4%

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

   8. sqr-neg 94.4%

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

   9. sqrt-unprod 50.2%

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

   10. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in eh around 0 99.1%

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

7. Final simplification 99.1%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. associate-/l* 99.6%

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

   5. hypot-1-def 99.6%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.4%

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

   8. sqr-neg 94.4%

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

   9. sqrt-unprod 50.2%

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

   10. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in eh around 0 99.1%

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

   1. expm1-log1p-u 88.2%

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

   2. expm1-udef 77.8%

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

   3. associate-*r* 77.8%

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

   4. *-commutative 77.8%

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

   5. add-sqr-sqrt 40.3%

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

   6. sqrt-unprod 79.9%

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

   7. sqr-neg 79.9%

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

   8. sqrt-unprod 38.9%

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

   9. add-sqr-sqrt 72.9%

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

8. Applied egg-rr 72.9%

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

   1. expm1-def 83.4%

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

   2. expm1-log1p 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*r/ 99.1%

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

   5. associate-/l* 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 6: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan(((eh / ew) * -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[(N[(eh / ew), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. associate-/l* 99.6%

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

   5. hypot-1-def 99.6%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.4%

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

   8. sqr-neg 94.4%

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

   9. sqrt-unprod 50.2%

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

   10. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in eh around 0 99.1%

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

7. Taylor expanded in t around 0 98.9%

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

   1. mul-1-neg 79.3%

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

   2. associate-/l* 79.3%

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

   3. associate-/r/ 79.3%

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

   4. distribute-rgt-neg-in 79.3%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 7: 78.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. associate-/l* 99.6%

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

   5. hypot-1-def 99.6%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.4%

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

   8. sqr-neg 94.4%

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

   9. sqrt-unprod 50.2%

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

   10. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in eh around 0 99.1%

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

7. Taylor expanded in t around 0 79.3%

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

8. Final simplification 79.3%

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

Alternative 8: 78.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. associate-/l* 99.6%

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

   5. hypot-1-def 99.6%

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

   6. add-sqr-sqrt 49.5%

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

   7. sqrt-unprod 94.4%

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

   8. sqr-neg 94.4%

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

   9. sqrt-unprod 50.2%

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

   10. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in eh around 0 99.1%

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

7. Taylor expanded in t around 0 79.3%

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

8. Taylor expanded in t around 0 79.3%

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

   1. mul-1-neg 79.3%

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

   2. associate-/l* 79.3%

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

   3. associate-/r/ 79.3%

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

   4. distribute-rgt-neg-in 79.3%

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

10. Simplified 79.3%

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

11. Final simplification 79.3%

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

Reproduce

herbie shell --seed 2023336 
(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)))))))