Trigonometry A

?

Percentage Accurate: 99.8% → 99.8%
Time: 10.6s
Precision: binary64
Cost: 19648

?

\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[\frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \sin v \]
(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)))
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);
}

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

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]

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

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Derivation?

  1. Initial program 99.8%

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

  2. Simplified99.8%

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

    [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 simplification99.8%

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

Alternatives

Alternative 1
Accuracy99.8%
Cost13376
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
Alternative 2
Accuracy99.6%
Cost13248
\[\frac{\sin v}{\cos v + \frac{1}{e}} \]
Alternative 3
Accuracy98.7%
Cost6848
\[\frac{e \cdot \sin v}{e + 1} \]
Alternative 4
Accuracy97.6%
Cost6592
\[e \cdot \sin v \]
Alternative 5
Accuracy52.1%
Cost1216
\[\frac{e}{\left(v \cdot 0.16666666666666666 + -0.3333333333333333 \cdot \left(e \cdot v\right)\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]
Alternative 6
Accuracy51.7%
Cost832
\[\frac{e}{-0.3333333333333333 \cdot \left(e \cdot v\right) + \frac{e + 1}{v}} \]
Alternative 7
Accuracy52.1%
Cost832
\[\frac{e}{v \cdot 0.16666666666666666 + \left(\frac{e}{v} + \frac{1}{v}\right)} \]
Alternative 8
Accuracy51.0%
Cost576
\[\frac{e}{v \cdot 0.16666666666666666 + \frac{1}{v}} \]
Alternative 9
Accuracy50.5%
Cost448
\[v \cdot \left(e - e \cdot e\right) \]
Alternative 10
Accuracy51.1%
Cost448
\[e \cdot \frac{v}{e + 1} \]
Alternative 11
Accuracy50.0%
Cost192
\[e \cdot v \]
Alternative 12
Accuracy4.5%
Cost64
\[v \]

Error

Reproduce?

herbie shell --seed 2023160 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))