Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 18.0s

Precision: binary64

Cost: 116864

Math

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

↓

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

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 (log (sqrt (exp (cos (atan (* (/ eh ew) (tan t)))))))))
   (fabs
    (-
     (* (+ t_1 t_1) (* ew (cos t)))
     (* (* eh (sin t)) (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) {
	double t_1 = log(sqrt(exp(cos(atan(((eh / ew) * tan(t)))))));
	return fabs((((t_1 + t_1) * (ew * cos(t))) - ((eh * sin(t)) * 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
    real(8) :: t_1
    t_1 = log(sqrt(exp(cos(atan(((eh / ew) * tan(t)))))))
    code = abs((((t_1 + t_1) * (ew * cos(t))) - ((eh * sin(t)) * 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) {
	double t_1 = Math.log(Math.sqrt(Math.exp(Math.cos(Math.atan(((eh / ew) * Math.tan(t)))))));
	return Math.abs((((t_1 + t_1) * (ew * Math.cos(t))) - ((eh * Math.sin(t)) * 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):
	t_1 = math.log(math.sqrt(math.exp(math.cos(math.atan(((eh / ew) * math.tan(t)))))))
	return math.fabs((((t_1 + t_1) * (ew * math.cos(t))) - ((eh * math.sin(t)) * 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)
	t_1 = log(sqrt(exp(cos(atan(Float64(Float64(eh / ew) * tan(t)))))))
	return abs(Float64(Float64(Float64(t_1 + t_1) * Float64(ew * cos(t))) - Float64(Float64(eh * sin(t)) * 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)
	t_1 = log(sqrt(exp(cos(atan(((eh / ew) * tan(t)))))));
	tmp = abs((((t_1 + t_1) * (ew * cos(t))) - ((eh * sin(t)) * 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_] := Block[{t$95$1 = N[Log[N[Sqrt[N[Exp[N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(t$95$1 + t$95$1), $MachinePrecision] * N[(ew * N[Cos[t], $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]]

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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	58880

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

Alternative 2
Error	0.1
Cost	52544

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

Alternative 3
Error	0.7
Cost	46144

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

Alternative 4
Error	0.6
Cost	46144

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

Alternative 5
Error	1.0
Cost	45632

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

Alternative 6
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 7
Error	1.2
Cost	32896

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

Alternative 8
Error	13.7
Cost	32768

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

Alternative 9
Error	13.7
Cost	32704

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

Alternative 10
Error	13.7
Cost	26368

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

Reproduce

herbie shell --seed 2022308 
(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)))))))