Trigonometry A

Average Error: 0.1 → 0.1

Time: 10.2s

Precision: binary64

Cost: 13376

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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

MATLAB

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   associate-*r/ [<=] 0.1	\[ \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]
   *-commutative [<=] 0.1	\[ \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]
   +-commutative [=>] 0.1	\[ \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]
   fma-def [=>] 0.1	\[ \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

3. Taylor expanded in v around inf 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.8
Cost	7232

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

Alternative 2
Error	1.1
Cost	6848

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

Alternative 3
Error	0.9
Cost	6848

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

Alternative 4
Error	1.4
Cost	6592

\[\sin v \cdot e \]

Alternative 5
Error	30.1
Cost	1344

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

Alternative 6
Error	30.7
Cost	576

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

Alternative 7
Error	30.8
Cost	448

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

Alternative 8
Error	31.4
Cost	192

\[v \cdot e \]

Alternative 9
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023056 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))