Trigonometry A

Average Error: 0.1 → 0.1

Time: 8.3s

Precision: binary64

Cost: 13504

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

Math

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

↓

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

FPCore

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

↓

(FPCore (e v)
 :precision binary64
 (* (/ 1.0 (+ 1.0 (* (cos v) e))) (* e (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 (1.0 / (1.0 + (cos(v) * e))) * (e * 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 = (1.0d0 / (1.0d0 + (cos(v) * e))) * (e * 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 (1.0 / (1.0 + (Math.cos(v) * e))) * (e * Math.sin(v));
}

Python

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

↓

def code(e, v):
	return (1.0 / (1.0 + (math.cos(v) * e))) * (e * 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(1.0 / Float64(1.0 + Float64(cos(v) * e))) * Float64(e * 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 = (1.0 / (1.0 + (cos(v) * e))) * (e * 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[(1.0 / N[(1.0 + N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.3

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

   [Start] 0.1	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   associate-/l* [=>] 0.3	\[ \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}} \]

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \left(e \cdot \sin v\right)} \]

4. Taylor expanded in v around inf 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	13376

\[\sin v \cdot \left(e - e \cdot \left(\cos v \cdot e\right)\right) \]

Alternative 2
Error	0.3
Cost	13376

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

Alternative 3
Error	0.1
Cost	13376

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

Alternative 4
Error	1.4
Cost	6592

\[e \cdot \sin v \]

Alternative 5
Error	30.4
Cost	1344

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

Alternative 6
Error	31.4
Cost	448

\[v \cdot \left(e - e \cdot e\right) \]

Alternative 7
Error	31.1
Cost	448

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

Alternative 8
Error	31.1
Cost	448

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

Alternative 9
Error	31.8
Cost	192

\[v \cdot e \]

Alternative 10
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023002 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))