Average Error: 15.1 → 0.3
Time: 15.4s
Precision: binary64
Cost: 32704
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot r \]
(FPCore (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))
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;
}

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

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;
}

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

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

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

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]

\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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.1

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

  2. Simplified15.1

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
    Proof
    (/.f64 (*.f64 r (sin.f64 b)) (cos.f64 (+.f64 b a))): 0 points increase in error, 0 points decrease in error
    (/.f64 (*.f64 r (sin.f64 b)) (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 a b)))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.3

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

  4. Taylor expanded in r around 0 0.3

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

  5. Simplified0.3

    \[\leadsto \color{blue}{\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b} \]
    Proof
    (*.f64 (/.f64 r (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 b) (sin.f64 a)))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 r (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 a) (sin.f64 b))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 r (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (neg.f64 (*.f64 (sin.f64 a) (sin.f64 b)))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 r (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b)))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 r (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r/_binary64 (/.f64 r (/.f64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))) (sin.f64 b)))): 49 points increase in error, 31 points decrease in error
    (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 r (sin.f64 b)) (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))): 25 points increase in error, 46 points decrease in error
    (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 b) r)) (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error

  6. Taylor expanded in r around 0 0.3

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

  7. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot r} \]
    Proof
    (*.f64 (/.f64 (sin.f64 b) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 b) (sin.f64 a)))) r): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 b) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 a) (sin.f64 b))))) r): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sin.f64 b) r) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 a) (sin.f64 b))))): 37 points increase in error, 25 points decrease in error

  8. Final simplification0.3

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

Alternatives

Alternative 1
Error15.2
Cost13384
\[\begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos b}\\ \mathbf{if}\;b \leq -1.6953668496300553 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.010832887328943094:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error15.2
Cost13248
\[\frac{\sin b}{\frac{\cos \left(b + a\right)}{r}} \]
Alternative 3
Error15.1
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 4
Error15.1
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 5
Error28.6
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 6
Error31.3
Cost6720
\[\frac{r}{\frac{\cos a}{b}} \]
Alternative 7
Error31.3
Cost6720
\[\frac{b \cdot r}{\cos a} \]
Alternative 8
Error31.3
Cost6720
\[r \cdot \frac{b}{\cos a} \]
Alternative 9
Error42.0
Cost192
\[b \cdot r \]

Error

Reproduce

herbie shell --seed 2022298 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))