Example 2 from Robby

Average Accuracy: 99.8% → 99.8%

Time: 29.9s

Precision: binary64

Cost: 71808

Math

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

↓

\[\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right) - \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)} + -1\right) \cdot \left(ew \cdot \cos t\right)\right| \]

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
  (-
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) (- eh)) ew))))
   (*
    (+ (exp (log1p (cos (atan (* eh (/ (tan t) ew)))))) -1.0)
    (* ew (cos t))))))

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((((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew)))) - ((exp(log1p(cos(atan((eh * (tan(t) / ew)))))) + -1.0) * (ew * cos(t)))));
}

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((((eh * Math.sin(t)) * Math.sin(Math.atan(((Math.tan(t) * -eh) / ew)))) - ((Math.exp(Math.log1p(Math.cos(Math.atan((eh * (Math.tan(t) / ew)))))) + -1.0) * (ew * Math.cos(t)))));
}

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((((eh * math.sin(t)) * math.sin(math.atan(((math.tan(t) * -eh) / ew)))) - ((math.exp(math.log1p(math.cos(math.atan((eh * (math.tan(t) / ew)))))) + -1.0) * (ew * math.cos(t)))))

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(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * Float64(-eh)) / ew)))) - Float64(Float64(exp(log1p(cos(atan(Float64(eh * Float64(tan(t) / ew)))))) + -1.0) * Float64(ew * cos(t)))))
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[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Exp[N[Log[1 + N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   Proof

   [Start] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   expm1-log1p-u [=>] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   expm1-udef [=>] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   div-inv [=>] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\left(\left(-eh\right) \cdot \tan t\right) \cdot \frac{1}{ew}\right)}\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   associate-*l* [=>] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \left(\tan t \cdot \frac{1}{ew}\right)\right)}\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   add-sqr-sqrt [=>] 50.2	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(\color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \left(\tan t \cdot \frac{1}{ew}\right)\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   sqrt-unprod [=>] 92.7	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \left(\tan t \cdot \frac{1}{ew}\right)\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   sqr-neg [=>] 92.7	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(\sqrt{\color{blue}{eh \cdot eh}} \cdot \left(\tan t \cdot \frac{1}{ew}\right)\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   sqrt-unprod [<=] 49.6	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(\color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \left(\tan t \cdot \frac{1}{ew}\right)\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   add-sqr-sqrt [<=] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(\color{blue}{eh} \cdot \left(\tan t \cdot \frac{1}{ew}\right)\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   associate-*r/ [=>] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(eh \cdot \color{blue}{\frac{\tan t \cdot 1}{ew}}\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   *-commutative [<=] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(eh \cdot \frac{\color{blue}{1 \cdot \tan t}}{ew}\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   *-un-lft-identity [<=] 99.8	\[ \left|\left(ew \cdot \cos t\right) \cdot \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(eh \cdot \frac{\color{blue}{\tan t}}{ew}\right)\right)} - 1\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

3. Final simplification 99.8%

   \[\leadsto \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right) - \left(e^{\mathsf{log1p}\left(\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)} + -1\right) \cdot \left(ew \cdot \cos t\right)\right| \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	52544

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

Alternative 2
Accuracy	99.8%
Cost	52544

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

Alternative 3
Accuracy	99.0%
Cost	46144

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

Alternative 4
Accuracy	98.5%
Cost	39296

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

Alternative 5
Accuracy	98.3%
Cost	32896

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

Alternative 6
Accuracy	79.8%
Cost	32768

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

Alternative 7
Accuracy	79.8%
Cost	26368

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

Alternative 8
Accuracy	79.5%
Cost	13120

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

Alternative 9
Accuracy	79.5%
Cost	13120

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

Alternative 10
Accuracy	43.2%
Cost	6464

\[\left|ew\right| \]

Reproduce

herbie shell --seed 2023131 
(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)))))))