Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 20.6s

Precision: binary64

Cost: 91840

Math

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

↓

\[\begin{array}{l} t_1 := \frac{eh \cdot \tan t}{ew}\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\\ \left|\frac{\sqrt{\cos t_2}}{\sqrt{\sqrt{1 + t_1 \cdot t_1}}} \cdot \left(ew \cdot \cos t\right) - \left(eh \cdot \sin t\right) \cdot \sin t_2\right| \end{array} \]

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
 (let* ((t_1 (/ (* eh (tan t)) ew)) (t_2 (atan (/ (* eh (- (tan t))) ew))))
   (fabs
    (-
     (* (/ (sqrt (cos t_2)) (sqrt (sqrt (+ 1.0 (* t_1 t_1))))) (* ew (cos t)))
     (* (* eh (sin t)) (sin t_2))))))

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) {
	double t_1 = (eh * tan(t)) / ew;
	double t_2 = atan(((eh * -tan(t)) / ew));
	return fabs((((sqrt(cos(t_2)) / sqrt(sqrt((1.0 + (t_1 * t_1))))) * (ew * cos(t))) - ((eh * sin(t)) * sin(t_2))));
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (eh * tan(t)) / ew
    t_2 = atan(((eh * -tan(t)) / ew))
    code = abs((((sqrt(cos(t_2)) / sqrt(sqrt((1.0d0 + (t_1 * t_1))))) * (ew * cos(t))) - ((eh * sin(t)) * sin(t_2))))
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) {
	double t_1 = (eh * Math.tan(t)) / ew;
	double t_2 = Math.atan(((eh * -Math.tan(t)) / ew));
	return Math.abs((((Math.sqrt(Math.cos(t_2)) / Math.sqrt(Math.sqrt((1.0 + (t_1 * t_1))))) * (ew * Math.cos(t))) - ((eh * Math.sin(t)) * Math.sin(t_2))));
}

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):
	t_1 = (eh * math.tan(t)) / ew
	t_2 = math.atan(((eh * -math.tan(t)) / ew))
	return math.fabs((((math.sqrt(math.cos(t_2)) / math.sqrt(math.sqrt((1.0 + (t_1 * t_1))))) * (ew * math.cos(t))) - ((eh * math.sin(t)) * math.sin(t_2))))

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)
	t_1 = Float64(Float64(eh * tan(t)) / ew)
	t_2 = atan(Float64(Float64(eh * Float64(-tan(t))) / ew))
	return abs(Float64(Float64(Float64(sqrt(cos(t_2)) / sqrt(sqrt(Float64(1.0 + Float64(t_1 * t_1))))) * Float64(ew * cos(t))) - Float64(Float64(eh * sin(t)) * sin(t_2))))
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)
	t_1 = (eh * tan(t)) / ew;
	t_2 = atan(((eh * -tan(t)) / ew));
	tmp = abs((((sqrt(cos(t_2)) / sqrt(sqrt((1.0 + (t_1 * t_1))))) * (ew * cos(t))) - ((eh * sin(t)) * sin(t_2))));
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_] := Block[{t$95$1 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(N[Sqrt[N[Cos[t$95$2], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	85056

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

Alternative 2
Error	0.1
Cost	58944

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

Alternative 3
Error	0.2
Cost	52736

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

Alternative 4
Error	0.2
Cost	52608

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

Alternative 5
Error	0.6
Cost	52544

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

Alternative 6
Error	0.7
Cost	46336

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

Alternative 7
Error	0.7
Cost	46208

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

Alternative 8
Error	1.0
Cost	46144

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

Alternative 9
Error	1.0
Cost	39296

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

Alternative 10
Error	1.2
Cost	32896

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

Alternative 11
Error	1.2
Cost	32832

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

Alternative 12
Error	13.4
Cost	32704

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

Alternative 13
Error	13.4
Cost	26368

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

Alternative 14
Error	13.6
Cost	13120

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

Alternative 15
Error	36.6
Cost	6464

\[\left|ew\right| \]

Reproduce

herbie shell --seed 2023010 
(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)))))))