Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 13.0s

Precision: binary64

Cost: 58944

Math

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

↓

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

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 (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) {
	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 = 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
    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
    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) {
	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 = 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):
	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 = 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)
	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 = atan(Float64(Float64(eh * Float64(-tan(t))) / ew))
	return abs(Float64(Float64(ew * Float64(cos(t) * cos(t_1))) - Float64(Float64(eh * sin(t)) * sin(t_1))))
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 = 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_] := 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[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $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. Simplified 0.1

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

   [Start] 0.1	\[ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	52544

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

Alternative 2
Error	6.4
Cost	46144

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

Alternative 3
Error	13.5
Cost	46016

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

Alternative 4
Error	14.2
Cost	39616

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

Alternative 5
Error	29.5
Cost	33216

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

Reproduce

herbie shell --seed 2023092 
(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)))))))