Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 17.1s

Precision: binary64

Cost: 58880

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

↓

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

↓

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

↓

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

↓

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

MATLAB

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

↓

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * cos(atan(((eh / ew) * tan(t))))) - ((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[Cos[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	52544

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

Alternative 2
Error	0.8
Cost	52480

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

Alternative 3
Error	0.7
Cost	46144

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

Alternative 4
Error	0.9
Cost	46080

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

Alternative 5
Error	6.5
Cost	39744

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

Alternative 6
Error	6.7
Cost	39680

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

Alternative 7
Error	13.0
Cost	39616

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

Alternative 8
Error	13.3
Cost	39552

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

Alternative 9
Error	13.5
Cost	32704

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

Alternative 10
Error	13.5
Cost	26368

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

Reproduce

herbie shell --seed 2022317 
(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)))))))