Trigonometry A

Average Error: 0.1 → 0.1

Time: 8.6s

Precision: binary64

Cost: 19648

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

Math

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

↓

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

FPCore

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

↓

(FPCore (e v) :precision binary64 (* (/ e (fma e (cos v) 1.0)) (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 / fma(e, cos(v), 1.0)) * 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 / fma(e, cos(v), 1.0)) * 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[(e * N[Cos[v], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[v], $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
   (/.f64 e (/.f64 (+.f64 1 (*.f64 e (cos.f64 v))) (sin.f64 v))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 e (sin.f64 v)) (+.f64 1 (*.f64 e (cos.f64 v))))): 13 points increase in error, 40 points decrease in error

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	0.3
Cost	13248

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

Reproduce

herbie shell --seed 2022334 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))