rsin A (should all be same)

Average Accuracy: 75.3% → 99.5%

Time: 18.2s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

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 (sin(b) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (r * sin(b)) / cos((a + b))
end function

↓

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (sin(b) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r
end function

Java

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

↓

public static double code(double r, double a, double b) {
	return (Math.sin(b) / ((Math.cos(a) * Math.cos(b)) - (Math.sin(b) * Math.sin(a)))) * r;
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / math.cos((a + b))

↓

def code(r, a, b):
	return (math.sin(b) / ((math.cos(a) * math.cos(b)) - (math.sin(b) * math.sin(a)))) * r

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(sin(b) / Float64(Float64(cos(a) * cos(b)) - Float64(sin(b) * sin(a)))) * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / cos((a + b));
end

↓

function tmp = code(r, a, b)
	tmp = (sin(b) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r;
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[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 75.3%

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

2. Applied egg-rr 99.5%

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

   [Start] 75.3	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   cos-sum [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}} \]
   cancel-sign-sub-inv [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b + \left(-\sin a\right) \cdot \sin b}} \]

3. Simplified 99.5%

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

   [Start] 99.5	\[ \frac{r \cdot \sin b}{\cos a \cdot \cos b + \left(-\sin a\right) \cdot \sin b} \]
   +-commutative [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\left(-\sin a\right) \cdot \sin b + \cos a \cdot \cos b}} \]
   *-commutative [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\sin b \cdot \left(-\sin a\right)} + \cos a \cdot \cos b} \]
   fma-def [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}} \]
   *-commutative [=>] 99.5	\[ \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin b, -\sin a, \color{blue}{\cos b \cdot \cos a}\right)} \]

4. Taylor expanded in r around 0 99.5%

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}} \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot r} \]
   Proof

   [Start] 99.5	\[ \frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \]
   associate-/l* [=>] 99.4	\[ \color{blue}{\frac{\sin b}{\frac{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}{r}}} \]
   associate-/r/ [=>] 99.5	\[ \color{blue}{\frac{\sin b}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \cdot r} \]
   +-commutative [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b + -1 \cdot \left(\sin a \cdot \sin b\right)}} \cdot r \]
   *-commutative [<=] 99.5	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} + -1 \cdot \left(\sin a \cdot \sin b\right)} \cdot r \]
   *-commutative [<=] 99.5	\[ \frac{\sin b}{\cos b \cdot \cos a + -1 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)}} \cdot r \]
   mul-1-neg [=>] 99.5	\[ \frac{\sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b \cdot \sin a\right)}} \cdot r \]
   unsub-neg [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]
   *-commutative [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b} - \sin b \cdot \sin a} \cdot r \]

6. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	32704

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

Alternative 2
Accuracy	99.4%
Cost	32512

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

Alternative 3
Accuracy	75.0%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-5} \lor \neg \left(b \leq 0.04\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 4
Accuracy	75.0%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-5} \lor \neg \left(b \leq 0.04\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 5
Accuracy	75.1%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -0.00035:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6
Accuracy	75.3%
Cost	13248

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

Alternative 7
Accuracy	75.3%
Cost	13248

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

Alternative 8
Accuracy	75.2%
Cost	13248

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

Alternative 9
Accuracy	53.1%
Cost	13120

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

Alternative 10
Accuracy	53.4%
Cost	7628

\[\begin{array}{l} t_0 := \sin b \cdot r\\ t_1 := \cos \left(b + a\right)\\ \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{b \cdot r}{t_1}\\ \mathbf{elif}\;b \leq 15500000:\\ \;\;\;\;\frac{r}{t_1 \cdot \left(b \cdot 0.16666666666666666 + \frac{1}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	53.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \lor \neg \left(b \leq 30\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 12
Accuracy	53.4%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \lor \neg \left(b \leq 17\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 13
Accuracy	53.4%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 17\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 14
Accuracy	37.3%
Cost	6592

\[\sin b \cdot r \]

Alternative 15
Accuracy	32.6%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023135 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))