rsin A (should all be same)

Average Error: 15.0 → 0.3

Time: 15.1s

Precision: binary64

Cost: 39040

Math

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

↓

\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \left(-\sin a\right)\right)} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))

↓

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos a) (cos b) (* (sin b) (- (sin a))))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + b));
}

↓

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(cos(a), cos(b), (sin(b) * -sin(a)));
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end

↓

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(a), cos(b), Float64(sin(b) * Float64(-sin(a)))))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.0

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

2. Applied egg-rr 0.3

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)}} \]

3. Final simplification 0.3

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \left(-\sin a\right)\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	19776

\[r \cdot \frac{1}{\frac{\cos a}{\tan b} - \sin a} \]

Alternative 2
Error	0.4
Cost	19648

\[\frac{r}{\frac{\cos a}{\tan b} - \sin a} \]

Alternative 3
Error	15.1
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00038 \lor \neg \left(a \leq 2.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 4
Error	15.0
Cost	13248

\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]

Alternative 5
Error	15.0
Cost	13248

\[\frac{r \cdot \sin b}{\cos \left(b - a\right)} \]

Alternative 6
Error	15.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-6} \lor \neg \left(b \leq 0.0065\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 7
Error	15.0
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-8} \lor \neg \left(b \leq 0.0065\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-8} \lor \neg \left(b \leq 0.0065\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 9
Error	25.6
Cost	6592

\[r \cdot \tan b \]

Alternative 10
Error	42.6
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023083 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))