rsin B

Average Error: 14.7 → 0.3

Time: 14.6s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

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

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 = (r / (cos(a) - (sin(a) * (sin(b) / cos(b))))) * tan(b)
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 (r / (Math.cos(a) - (Math.sin(a) * (Math.sin(b) / Math.cos(b))))) * Math.tan(b);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.7

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

2. Applied cos-sum_binary64 0.3

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

3. Taylor expanded in r around 0 0.3

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

4. Simplified 0.4

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \frac{\sin a}{1}}} \]

5. Applied frac-sub_binary64 0.4

   \[\leadsto \frac{r}{\color{blue}{\frac{\cos a \cdot 1 - \frac{\sin b}{\cos b} \cdot \sin a}{\frac{\sin b}{\cos b} \cdot 1}}} \]

6. Applied associate-/r/_binary64 0.3

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

7. Simplified 0.3

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

8. Applied quot-tan_binary64 0.3

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

9. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022137 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))