rsin A (should all be same)

Average Error: 15.2 → 0.3

Time: 25.2s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (sin b) (- (sin a)) (* (cos b) (cos 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(sin(b), -sin(a), (cos(b) * cos(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(sin(b), Float64(-sin(a)), Float64(cos(b) * cos(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[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.2

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

2. Applied egg-rr 0.3

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

3. Applied egg-rr 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	39040

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

Alternative 2
Error	0.3
Cost	32704

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

Alternative 3
Error	16.0
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	13248

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

Alternative 5
Error	15.2
Cost	13248

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

Alternative 6
Error	16.0
Cost	6984

\[\begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	25.4
Cost	6592

\[\tan b \cdot r \]

Alternative 8
Error	42.0
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023010 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))