rsin A (should all be same)

Average Error: 15.1 → 0.3

Time: 16.4s

Precision: binary64

Cost: 77760

Math

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

↓

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

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)
   (fma (- (sin a)) (sin b) (fma (- (sin b)) (sin a) (* (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), fma(-sin(a), sin(b), fma(-sin(b), sin(a), (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), fma(Float64(-sin(a)), sin(b), fma(Float64(-sin(b)), sin(a), Float64(sin(b) * 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[a], $MachinePrecision]) * N[Sin[b], $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.1

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

2. Applied egg-rr 0.3

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

3. Simplified 0.3

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, \mathsf{fma}\left(-\sin a, \sin b, \mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)\right)\right)}} \]
   Proof
   (/.f64 (*.f64 r (sin.f64 b)) (fma.f64 (cos.f64 a) (cos.f64 b) (fma.f64 (neg.f64 (sin.f64 a)) (sin.f64 b) (fma.f64 (neg.f64 (sin.f64 b)) (sin.f64 a) (*.f64 (sin.f64 b) (sin.f64 a)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 r (sin.f64 b)) (fma.f64 (cos.f64 a) (cos.f64 b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b)) (fma.f64 (neg.f64 (sin.f64 b)) (sin.f64 a) (*.f64 (sin.f64 b) (sin.f64 a))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 r (sin.f64 b)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (+.f64 (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b)) (fma.f64 (neg.f64 (sin.f64 b)) (sin.f64 a) (*.f64 (sin.f64 b) (sin.f64 a))))))): 0 points increase in error, 3 points decrease in error

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	45568

\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}} \]

Alternative 2
Error	0.3
Cost	32704

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

Alternative 3
Error	0.4
Cost	32512

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

Alternative 4
Error	15.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0027 \lor \neg \left(a \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 5
Error	15.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00155 \lor \neg \left(a \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00162:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 7
Error	15.3
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -0.0265:\\ \;\;\;\;\frac{t_0}{\cos a}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	13248

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

Alternative 9
Error	29.0
Cost	13120

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

Alternative 10
Error	30.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a \cdot \left(b \cdot 0.16666666666666666 + \frac{1}{b}\right)}\\ \end{array} \]

Alternative 11
Error	29.0
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+50} \lor \neg \left(b \leq 1.25\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 12
Error	29.1
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+50} \lor \neg \left(b \leq 1.45\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 13
Error	39.5
Cost	6592

\[r \cdot \sin b \]

Alternative 14
Error	42.3
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2022343 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))