Average Error: 0.1 → 0.1
Time: 6.8s
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

Error

Derivation

  1. Initial program 0.1

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}} \]
    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))))): 7 points increase in error, 44 points decrease in error

  3. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Alternatives

Alternative 1
Error0.3
Cost13376
\[\frac{e}{\frac{e}{\tan v} + \frac{1}{\sin v}} \]
Alternative 2
Error0.1
Cost13376
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
Alternative 3
Error1.1
Cost6848
\[\sin v \cdot \left(e - e \cdot e\right) \]
Alternative 4
Error0.9
Cost6848
\[\frac{\sin v}{\frac{e + 1}{e}} \]
Alternative 5
Error0.8
Cost6848
\[\frac{e \cdot \sin v}{e + 1} \]
Alternative 6
Error1.4
Cost6592
\[e \cdot \sin v \]
Alternative 7
Error30.9
Cost1088
\[\frac{e}{\frac{e}{v} + \left(\frac{1}{v} + v \cdot \left(0.16666666666666666 + e \cdot -0.3333333333333333\right)\right)} \]
Alternative 8
Error31.5
Cost576
\[\frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666} \]
Alternative 9
Error31.9
Cost448
\[e \cdot \left(v \cdot \left(1 - e\right)\right) \]
Alternative 10
Error31.9
Cost448
\[v \cdot \left(e - e \cdot e\right) \]
Alternative 11
Error31.6
Cost448
\[v \cdot \frac{e}{e + 1} \]
Alternative 12
Error32.2
Cost192
\[e \cdot v \]
Alternative 13
Error61.2
Cost64
\[v \]

Error

Reproduce

herbie shell --seed 2022332 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))