Example 2 from Robby

Average Error: 0.1 → 0.1

Time: 49.5s

Precision: binary64

Cost: 58944

Math

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

↓

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

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 (- (* (cos t) (* ew (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(((cos(t) * (ew * 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(((cos(t) * (ew * 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(((Math.cos(t) * (ew * 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(((math.cos(t) * (ew * 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(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(cos(t) * Float64(ew * 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(((cos(t) * (ew * 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[(N[Cos[t], $MachinePrecision] * N[(ew * N[Cos[t$95$1], $MachinePrecision]), $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

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

   [Start] 0.1	\[ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   rational_best-simplify-1 [=>] 0.1	\[ \left|\color{blue}{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \left(ew \cdot \cos t\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   rational_best-simplify-50 [=>] 0.1	\[ \left|\color{blue}{\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   rational_best-simplify-1 [=>] 0.1	\[ \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \color{blue}{\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)}\right| \]
   rational_best-simplify-1 [=>] 0.1	\[ \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \color{blue}{\left(\sin t \cdot eh\right)}\right| \]
   rational_best-simplify-50 [=>] 0.1	\[ \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	52544

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

Alternative 2
Error	10.4
Cost	46664

\[\begin{array}{l} t_1 := \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \sin \tan^{-1} \left(-ew \cdot \left(t \cdot \frac{\frac{eh}{ew}}{ew}\right)\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{if}\;ew \leq -1.7 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ew \leq 7.8 \cdot 10^{+86}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.3
Cost	46664

\[\begin{array}{l} t_1 := \cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\\ \mathbf{if}\;ew \leq -9 \cdot 10^{+63}:\\ \;\;\;\;\left|t_1 - \sin \tan^{-1} \left(-ew \cdot \left(t \cdot \frac{\frac{eh}{ew}}{ew}\right)\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 1.3 \cdot 10^{+88}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 - \sin \tan^{-1} \left(-ew \cdot \frac{t \cdot eh}{ew \cdot ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]

Alternative 4
Error	10.8
Cost	46280

\[\begin{array}{l} t_1 := \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)\\ t_2 := \sin t_1\\ t_3 := \left|\cos t \cdot \left(ew \cdot \cos t_1\right) - t_2 \cdot \left(t \cdot eh\right)\right|\\ \mathbf{if}\;ew \leq -4.8 \cdot 10^{+63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;ew \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) - eh \cdot \left(\sin t \cdot t_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	11.3
Cost	40008

\[\begin{array}{l} t_1 := \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)\\ t_2 := \sin t_1\\ t_3 := ew \cdot \cos t_1\\ t_4 := \left|\cos t \cdot t_3 - t_2 \cdot \left(t \cdot eh\right)\right|\\ \mathbf{if}\;ew \leq -7.8 \cdot 10^{+62}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;ew \leq 7.2 \cdot 10^{+86}:\\ \;\;\;\;\left|t_3 - eh \cdot \left(\sin t \cdot t_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 6
Error	14.0
Cost	39616

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

Alternative 7
Error	29.9
Cost	33216

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

Reproduce

herbie shell --seed 2023099 
(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)))))))