rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 12.9s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + 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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + 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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\mathsf{fma}\left(\sin \left(-b\right), \sin a, \cos b \cdot \cos a\right)} \cdot \sin b \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return (r / fma(sin(-b), sin(a), (cos(b) * cos(a)))) * sin(b);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.7

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

4. Applied rewrites 78.7%

   \[\leadsto \color{blue}{\frac{r}{\cos \left(b + a\right)} \cdot \sin b} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. unsub-neg N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-neg-in N/A

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

   15. lift-sin.f64 N/A

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

   16. sin-neg N/A

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

   17. lift-neg.f64 N/A

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

   18. lift-sin.f64 N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)} \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return sin(b) * (r / fma(cos(b), cos(a), -(sin(a) * sin(b))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 76.6%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 76.6

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 76.6

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

4. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\frac{r}{\cos \left(b + a\right)} \cdot \sin b} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. unsub-neg N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))