Trigonometry A

Average Error: 0.1 → 0.2

Time: 32.3s

Precision: binary64

Cost: 13632

\[0 \leq e \land e \leq 1\]

Math

\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]

↓

\[\frac{\sin v \cdot \left(e \cdot \frac{-2}{-1 - \cos v \cdot e}\right)}{2} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

↓

(FPCore (e v)
 :precision binary64
 (/ (* (sin v) (* e (/ -2.0 (- -1.0 (* (cos v) e))))) 2.0))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

↓

double code(double e, double v) {
	return (sin(v) * (e * (-2.0 / (-1.0 - (cos(v) * e))))) / 2.0;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

↓

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (sin(v) * (e * ((-2.0d0) / ((-1.0d0) - (cos(v) * e))))) / 2.0d0
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

↓

public static double code(double e, double v) {
	return (Math.sin(v) * (e * (-2.0 / (-1.0 - (Math.cos(v) * e))))) / 2.0;
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

↓

def code(e, v):
	return (math.sin(v) * (e * (-2.0 / (-1.0 - (math.cos(v) * e))))) / 2.0

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

↓

function code(e, v)
	return Float64(Float64(sin(v) * Float64(e * Float64(-2.0 / Float64(-1.0 - Float64(cos(v) * e))))) / 2.0)
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

↓

function tmp = code(e, v)
	tmp = (sin(v) * (e * (-2.0 / (-1.0 - (cos(v) * e))))) / 2.0;
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * N[(e * N[(-2.0 / N[(-1.0 - N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\left(\sin v \cdot \left(e + e\right)\right) \cdot \left(\frac{-1}{-1 - e \cdot \cos v} - \frac{0.5}{e \cdot \cos v + 1}\right)} \]

3. Simplified 0.1

   \[\leadsto \color{blue}{\sin v \cdot \left(\left(e + e\right) \cdot \left(\frac{-1}{-1 - \cos v \cdot e} - \frac{0.5}{1 + \cos v \cdot e}\right)\right)} \]
   Proof

   [Start] 0.2	\[ \left(\sin v \cdot \left(e + e\right)\right) \cdot \left(\frac{-1}{-1 - e \cdot \cos v} - \frac{0.5}{e \cdot \cos v + 1}\right) \]
   rational_best-simplify-1 [=>] 0.2	\[ \color{blue}{\left(\frac{-1}{-1 - e \cdot \cos v} - \frac{0.5}{e \cdot \cos v + 1}\right) \cdot \left(\sin v \cdot \left(e + e\right)\right)} \]
   rational_best-simplify-1 [=>] 0.2	\[ \left(\frac{-1}{-1 - e \cdot \cos v} - \frac{0.5}{e \cdot \cos v + 1}\right) \cdot \color{blue}{\left(\left(e + e\right) \cdot \sin v\right)} \]
   rational_best-simplify-50 [=>] 0.1	\[ \color{blue}{\sin v \cdot \left(\left(e + e\right) \cdot \left(\frac{-1}{-1 - e \cdot \cos v} - \frac{0.5}{e \cdot \cos v + 1}\right)\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \sin v \cdot \left(\left(e + e\right) \cdot \left(\frac{-1}{-1 - \color{blue}{\cos v \cdot e}} - \frac{0.5}{e \cdot \cos v + 1}\right)\right) \]
   rational_best-simplify-3 [<=] 0.1	\[ \sin v \cdot \left(\left(e + e\right) \cdot \left(\frac{-1}{-1 - \cos v \cdot e} - \frac{0.5}{\color{blue}{1 + e \cdot \cos v}}\right)\right) \]
   rational_best-simplify-1 [=>] 0.1	\[ \sin v \cdot \left(\left(e + e\right) \cdot \left(\frac{-1}{-1 - \cos v \cdot e} - \frac{0.5}{1 + \color{blue}{\cos v \cdot e}}\right)\right) \]

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\sin v \cdot \left(e \cdot \frac{-2}{-1 - \cos v \cdot e}\right)}{2}} \]

5. Final simplification 0.2

   \[\leadsto \frac{\sin v \cdot \left(e \cdot \frac{-2}{-1 - \cos v \cdot e}\right)}{2} \]

Alternatives

Alternative 1
Error	0.1
Cost	13440

\[-\sin v \cdot \frac{e}{-1 - e \cdot \cos v} \]

Alternative 2
Error	0.1
Cost	13376

\[e \cdot \frac{\sin v}{1 + e \cdot \cos v} \]

Alternative 3
Error	0.9
Cost	6848

\[e \cdot \frac{\sin v}{e + 1} \]

Alternative 4
Error	1.6
Cost	6592

\[\sin v \cdot e \]

Alternative 5
Error	31.4
Cost	448

\[e \cdot \frac{v}{e + 1} \]

Alternative 6
Error	32.2
Cost	192

\[v \cdot e \]

Alternative 7
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023099 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))