Trigonometry A

Average Error: 0.1 → 0.1

Time: 7.4s

Precision: binary64

Cost: 26304

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

Math

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

↓

\[\frac{e \cdot \sin v}{1 + \log \left(1 + \mathsf{expm1}\left(e \cdot \cos v\right)\right)} \]

FPCore

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

↓

(FPCore (e v)
 :precision binary64
 (/ (* e (sin v)) (+ 1.0 (log (+ 1.0 (expm1 (* e (cos 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 * sin(v)) / (1.0 + log((1.0 + expm1((e * cos(v))))));
}

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.log((1.0 + Math.expm1((e * Math.cos(v))))));
}

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.log((1.0 + math.expm1((e * math.cos(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 * sin(v)) / Float64(1.0 + log(Float64(1.0 + expm1(Float64(e * cos(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[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Log[N[(1.0 + N[(Exp[N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

   \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(e \cdot \cos v\right)\right)}} \]

3. Final simplification 0.1

   \[\leadsto \frac{e \cdot \sin v}{1 + \log \left(1 + \mathsf{expm1}\left(e \cdot \cos v\right)\right)} \]

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	0.1
Cost	13376

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

Alternative 3
Error	0.2
Cost	13248

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

Alternative 4
Error	0.9
Cost	6848

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

Alternative 5
Error	1.4
Cost	6592

\[e \cdot \sin v \]

Alternative 6
Error	29.3
Cost	448

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

Alternative 7
Error	31.6
Cost	192

\[e \cdot v \]

Alternative 8
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2022338 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))