Trigonometry A

Average Error: 0.21% → 0.21%

Time: 9.5s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (e / ((-1.0d0) - (e * cos(v)))) * -sin(v)
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 (e / (-1.0 - (e * Math.cos(v)))) * -Math.sin(v);
}

Python

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

↓

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

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(e / Float64(-1.0 - Float64(e * cos(v)))) * Float64(-sin(v)))
end

MATLAB

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

↓

function tmp = code(e, v)
	tmp = (e / (-1.0 - (e * cos(v)))) * -sin(v);
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[(e / N[(-1.0 - N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sin[v], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 0.21

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

2. Applied egg-rr 0.41

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

3. Simplified 0.21

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

   [Start] 0.41	\[ -\frac{e}{\frac{-1 - e \cdot \cos v}{\sin v}} \]
   associate-/r/ [=>] 0.21	\[ -\color{blue}{\frac{e}{-1 - e \cdot \cos v} \cdot \sin v} \]
   distribute-rgt-neg-in [=>] 0.21	\[ \color{blue}{\frac{e}{-1 - e \cdot \cos v} \cdot \left(-\sin v\right)} \]

4. Final simplification 0.21

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

Alternatives

Alternative 1
Error	0.22%
Cost	13376

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

Alternative 2
Error	0.21%
Cost	13376

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

Alternative 3
Error	0.39%
Cost	13248

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

Alternative 4
Error	1.18%
Cost	6848

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

Alternative 5
Error	2.11%
Cost	6592

\[e \cdot \sin v \]

Alternative 6
Error	47.26%
Cost	1344

\[\frac{e}{v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]

Alternative 7
Error	48.18%
Cost	576

\[\frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666} \]

Alternative 8
Error	48.37%
Cost	448

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

Alternative 9
Error	49.29%
Cost	192

\[e \cdot v \]

Alternative 10
Error	95.5%
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023104 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))