Trigonometry A

Average Accuracy: 99.8% → 99.8%

Time: 10.8s

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 99.8%

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

2. Simplified 99.8%

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

   [Start] 99.8	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   associate-*l/ [<=] 99.8	\[ \color{blue}{\frac{e}{1 + e \cdot \cos v} \cdot \sin v} \]
   +-commutative [=>] 99.8	\[ \frac{e}{\color{blue}{e \cdot \cos v + 1}} \cdot \sin v \]
   fma-def [=>] 99.8	\[ \frac{e}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot \sin v \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13376

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

Alternative 2
Accuracy	99.6%
Cost	13248

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

Alternative 3
Accuracy	98.8%
Cost	6848

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

Alternative 4
Accuracy	97.6%
Cost	6592

\[e \cdot \sin v \]

Alternative 5
Accuracy	56.0%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;v \leq -2.1:\\ \;\;\;\;\frac{e}{\frac{1}{v} + -1}\\ \mathbf{elif}\;v \leq 1.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{e}{v \cdot \left(e \cdot -0.5 - -0.16666666666666666\right) + \left(\frac{1}{v} + \frac{e}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{1 + \frac{1}{v}}\\ \end{array} \]

Alternative 6
Accuracy	56.0%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;v \leq -2.1:\\ \;\;\;\;\frac{e}{\frac{1}{v} + -1}\\ \mathbf{elif}\;v \leq 1.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{e}{\left(\frac{1}{v} + \frac{e}{v}\right) + v \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{1 + \frac{1}{v}}\\ \end{array} \]

Alternative 7
Accuracy	55.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;v \leq -1.65:\\ \;\;\;\;\frac{e}{\frac{1}{v} + -1}\\ \mathbf{elif}\;v \leq 1.6:\\ \;\;\;\;\frac{e}{\frac{e + 1}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{1 + \frac{1}{v}}\\ \end{array} \]

Alternative 8
Accuracy	55.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;v \leq -1.6:\\ \;\;\;\;\frac{e}{\frac{1}{v} + -1}\\ \mathbf{elif}\;v \leq 1.6:\\ \;\;\;\;\frac{v}{1 + \frac{1}{e}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{1 + \frac{1}{v}}\\ \end{array} \]

Alternative 9
Accuracy	55.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;v \leq -1.65:\\ \;\;\;\;\frac{e}{\frac{1}{v} + -1}\\ \mathbf{elif}\;v \leq 1.6:\\ \;\;\;\;\frac{e \cdot v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{1 + \frac{1}{v}}\\ \end{array} \]

Alternative 10
Accuracy	51.0%
Cost	448

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

Alternative 11
Accuracy	54.1%
Cost	448

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

Alternative 12
Accuracy	54.2%
Cost	448

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

Alternative 13
Accuracy	50.5%
Cost	192

\[e \cdot v \]

Alternative 14
Accuracy	4.5%
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023151 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))