rsin A (should all be same)

Average Error: 15.5 → 0.4

Time: 1.0min

Precision: binary64

Cost: 45888

Math

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

↓

\[\sin b \cdot \frac{r}{\cos b \cdot \left(\cos a \cdot 2\right) - \left(\sin b \cdot \sin a + \cos b \cdot \cos a\right)} \]

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (*
  (sin b)
  (/
   r
   (-
    (* (cos b) (* (cos a) 2.0))
    (+ (* (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 sin(b) * (r / ((cos(b) * (cos(a) * 2.0)) - ((sin(b) * sin(a)) + (cos(b) * cos(a)))));
}

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) * (r / ((cos(b) * (cos(a) * 2.0d0)) - ((sin(b) * sin(a)) + (cos(b) * cos(a)))))
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) * (r / ((Math.cos(b) * (Math.cos(a) * 2.0)) - ((Math.sin(b) * Math.sin(a)) + (Math.cos(b) * Math.cos(a)))));
}

Python

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

↓

def code(r, a, b):
	return math.sin(b) * (r / ((math.cos(b) * (math.cos(a) * 2.0)) - ((math.sin(b) * math.sin(a)) + (math.cos(b) * math.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(sin(b) * Float64(r / Float64(Float64(cos(b) * Float64(cos(a) * 2.0)) - Float64(Float64(sin(b) * sin(a)) + Float64(cos(b) * cos(a))))))
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) * (r / ((cos(b) * (cos(a) * 2.0)) - ((sin(b) * sin(a)) + (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[Sin[b], $MachinePrecision] * N[(r / N[(N[(N[Cos[b], $MachinePrecision] * N[(N[Cos[a], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.5

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

2. Simplified 15.5

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

   [Start] 15.5	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   rational_best-simplify-3 [=>] 15.5	\[ \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Taylor expanded in r around 0 0.3

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

5. Simplified 15.5

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

   [Start] 0.3	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   rational_best-simplify-1 [=>] 0.3	\[ \frac{\color{blue}{r \cdot \sin b}}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   trig-simplify-14 [=>] 15.5	\[ \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]
   rational_best-simplify-3 [<=] 15.5	\[ \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]
   rational_best-simplify-8 [<=] 15.5	\[ \frac{r \cdot \sin b}{\color{blue}{\frac{\cos \left(b + a\right)}{1}}} \]
   metadata-eval [<=] 15.5	\[ \frac{r \cdot \sin b}{\frac{\cos \left(b + a\right)}{\color{blue}{-1 \cdot -1}}} \]
   rational_best-simplify-53 [<=] 15.5	\[ \frac{r \cdot \sin b}{\color{blue}{\frac{\frac{\cos \left(b + a\right)}{-1}}{-1}}} \]
   rational_best-simplify-13 [<=] 15.5	\[ \frac{r \cdot \sin b}{\frac{\color{blue}{-\cos \left(b + a\right)}}{-1}} \]
   rational_best-simplify-12 [=>] 15.5	\[ \frac{r \cdot \sin b}{\color{blue}{-\left(-\cos \left(b + a\right)\right)}} \]
   rational_best-simplify-61 [<=] 15.5	\[ \color{blue}{\left(-r\right) \cdot \frac{\sin b}{-\cos \left(b + a\right)}} \]
   rational_best-simplify-55 [=>] 15.5	\[ \color{blue}{\sin b \cdot \frac{-r}{-\cos \left(b + a\right)}} \]
   rational_best-simplify-14 [=>] 15.5	\[ \sin b \cdot \frac{\color{blue}{0 - r}}{-\cos \left(b + a\right)} \]
   rational_best-simplify-58 [=>] 15.5	\[ \sin b \cdot \color{blue}{\frac{r - 0}{\cos \left(b + a\right)}} \]
   rational_best-simplify-9 [=>] 15.5	\[ \sin b \cdot \frac{\color{blue}{r}}{\cos \left(b + a\right)} \]

6. Applied egg-rr 0.4

   \[\leadsto \sin b \cdot \frac{r}{\color{blue}{\cos b \cdot \left(\cos a \cdot 2\right) - \left(\sin b \cdot \sin a + \cos b \cdot \cos a\right)}} \]

7. Final simplification 0.4

   \[\leadsto \sin b \cdot \frac{r}{\cos b \cdot \left(\cos a \cdot 2\right) - \left(\sin b \cdot \sin a + \cos b \cdot \cos a\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	32704

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

Alternative 2
Error	14.9
Cost	26496

\[\frac{r \cdot \sin b}{\cos a \cdot \left(\cos b \cdot 2\right) + \left(-\cos \left(b - a\right)\right)} \]

Alternative 3
Error	15.6
Cost	13384

\[\begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{b}{\cos a} \cdot 1\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{b}{\cos a} \cdot 1\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 5
Error	15.5
Cost	13248

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

Alternative 6
Error	15.5
Cost	13248

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

Alternative 7
Error	29.2
Cost	13120

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

Alternative 8
Error	29.2
Cost	7112

\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -19000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{b}{\cos a} \cdot 1\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	29.2
Cost	6984

\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -19000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+21}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	39.2
Cost	6592

\[\sin b \cdot r \]

Alternative 11
Error	42.1
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023099 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))