Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 20.0s

Precision: binary64

Cost: 58944

Math

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

↓

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

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 (- (/ (tan t) ew))))))
   (* (sin t) (* eh (sin (atan (/ (* eh (- (tan t))) 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 * -(tan(t) / ew))))) - (sin(t) * (eh * sin(atan(((eh * -tan(t)) / 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 * -(tan(t) / ew))))) - (sin(t) * (eh * sin(atan(((eh * -tan(t)) / 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 * -(Math.tan(t) / ew))))) - (Math.sin(t) * (eh * Math.sin(Math.atan(((eh * -Math.tan(t)) / 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 * -(math.tan(t) / ew))))) - (math.sin(t) * (eh * math.sin(math.atan(((eh * -math.tan(t)) / 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(eh * Float64(-Float64(tan(t) / ew)))))) - Float64(sin(t) * Float64(eh * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / 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 * -(tan(t) / ew))))) - (sin(t) * (eh * sin(atan(((eh * -tan(t)) / 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[(eh * (-N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $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. 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. Taylor expanded in eh around 0 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	52672

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

Alternative 2
Error	0.1
Cost	52672

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

Alternative 3
Error	0.6
Cost	52544

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

Alternative 4
Error	0.9
Cost	39296

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

Alternative 5
Error	5.8
Cost	33161

\[\begin{array}{l} t_1 := ew \cdot \cos t\\ \mathbf{if}\;eh \leq -1.55 \cdot 10^{-174} \lor \neg \left(eh \leq 1.15 \cdot 10^{-81}\right):\\ \;\;\;\;\left|eh \cdot \sin t + t_1 \cdot \cos \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \cos \tan^{-1} \left(eh \cdot \left(-\frac{\tan t}{ew}\right)\right)\right|\\ \end{array} \]

Alternative 6
Error	5.8
Cost	33160

\[\begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := t_2 \cdot \cos \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\\ \mathbf{if}\;eh \leq -6.5 \cdot 10^{-174}:\\ \;\;\;\;\left|t_1 - t_3\right|\\ \mathbf{elif}\;eh \leq 9.5 \cdot 10^{-82}:\\ \;\;\;\;\left|t_2 \cdot \cos \tan^{-1} \left(eh \cdot \left(-\frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 + t_3\right|\\ \end{array} \]

Alternative 7
Error	8.9
Cost	33097

\[\begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -1.7 \cdot 10^{+71} \lor \neg \left(eh \leq 2.1 \cdot 10^{-15}\right):\\ \;\;\;\;\left|eh \cdot \sin t + ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-t_1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh + eh \cdot \left(-1 - t_1\right)\right)\right|\\ \end{array} \]

Alternative 8
Error	8.9
Cost	33033

\[\begin{array}{l} t_1 := \cos \tan^{-1} \left(eh \cdot \left(-\frac{\tan t}{ew}\right)\right)\\ \mathbf{if}\;eh \leq -1.4 \cdot 10^{+71} \lor \neg \left(eh \leq 2.05 \cdot 10^{-15}\right):\\ \;\;\;\;\left|eh \cdot \sin t + ew \cdot t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot t_1\right|\\ \end{array} \]

Alternative 9
Error	0.9
Cost	33024

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

Alternative 10
Error	24.7
Cost	32640

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

Alternative 11
Error	24.9
Cost	26368

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

Alternative 12
Error	30.1
Cost	26240

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

Alternative 13
Error	36.8
Cost	26048

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

Alternative 14
Error	37.0
Cost	19712

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

Alternative 15
Error	37.6
Cost	19648

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

Alternative 16
Error	38.1
Cost	13312

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

Reproduce

herbie shell --seed 2023059 
(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)))))))