rsin A

Average Error: 14.7 → 0.3

Time: 9.8s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (* (sin b) (/ (- r) (- (* (sin b) (sin a)) (* (cos a) (cos 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 sin(b) * (-r / ((sin(b) * sin(a)) - (cos(a) * cos(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 = sin(b) * (-r / ((sin(b) * sin(a)) - (cos(a) * cos(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 Math.sin(b) * (-r / ((Math.sin(b) * Math.sin(a)) - (Math.cos(a) * Math.cos(b))));
}

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.sin(b) * math.sin(a)) - (math.cos(a) * math.cos(b))))

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(Float64(-r) / Float64(Float64(sin(b) * sin(a)) - Float64(cos(a) * cos(b)))))
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 / ((sin(b) * sin(a)) - (cos(a) * cos(b))));
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[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.7

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

2. Applied cos-sum_binary64 0.3

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

3. Applied frac-2neg_binary64 0.3

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

4. Simplified 0.3

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

5. Simplified 0.3

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

6. Applied *-un-lft-identity_binary64 0.3

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

7. Applied distribute-rgt-neg-in_binary64 0.3

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

8. Applied times-frac_binary64 0.3

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

9. Simplified 0.3

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

10. Simplified 0.3

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

11. Final simplification 0.3

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

Reproduce

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