Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 19.8s

Precision: binary64

Cost: 58944

Math

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

↓

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

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

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

   [Start] 0.1	\[ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   fabs-sub [=>] 0.1	\[ \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]
   fabs-neg [<=] 0.1	\[ \color{blue}{\left|-\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right|} \]
   sub-neg [=>] 0.1	\[ \left|-\color{blue}{\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]
   +-commutative [=>] 0.1	\[ \left|-\color{blue}{\left(\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]
   distribute-neg-in [=>] 0.1	\[ \left|\color{blue}{\left(-\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	52736

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

Alternative 2
Error	0.7
Cost	52544

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

Alternative 3
Error	1.0
Cost	39296

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

Alternative 4
Error	23.9
Cost	32832

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

Alternative 5
Error	24.0
Cost	32640

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

Alternative 6
Error	24.2
Cost	26432

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

Alternative 7
Error	30.3
Cost	26240

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

Alternative 8
Error	36.8
Cost	26112

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

Alternative 9
Error	37.0
Cost	19904

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

Alternative 10
Error	37.0
Cost	19776

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

Alternative 11
Error	37.6
Cost	19648

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

Alternative 12
Error	38.2
Cost	13376

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

Alternative 13
Error	38.2
Cost	13376

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

Reproduce

herbie shell --seed 2023187 
(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)))))))