rsin A (should all be same)

Percentage Accurate: 76.9% → 99.4%

Time: 9.5s

Alternatives: 10

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: 76.9% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\mathsf{fma}\left(\sin a, -1, {\tan b}^{-1} \cdot \cos a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (fma (sin a) -1.0 (* (pow (tan b) -1.0) (cos a)))))

C

double code(double r, double a, double b) {
	return r / fma(sin(a), -1.0, (pow(tan(b), -1.0) * cos(a)));
}

Julia

function code(r, a, b)
	return Float64(r / fma(sin(a), -1.0, Float64((tan(b) ^ -1.0) * cos(a))))
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[Sin[a], $MachinePrecision] * -1.0 + N[(N[Power[N[Tan[b], $MachinePrecision], -1.0], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 77.7

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

4. Applied rewrites 77.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. div-sub N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{r}{\color{blue}{\frac{\cos a \cdot \cos b}{\sin b} - \frac{\sin a \cdot \sin b}{\sin b}}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-inverses N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.5

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

8. Applied rewrites 99.5%

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

Alternative 2: 50.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, b \cdot b, -0.019444444444444445\right), b \cdot b, -0.16666666666666666\right) \cdot b - {b}^{-1}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/
  r
  (*
   (-
    (*
     (fma
      (fma -0.00205026455026455 (* b b) -0.019444444444444445)
      (* b b)
      -0.16666666666666666)
     b)
    (pow b -1.0))
   (/ (cos (+ a b)) -1.0))))

C

double code(double r, double a, double b) {
	return r / (((fma(fma(-0.00205026455026455, (b * b), -0.019444444444444445), (b * b), -0.16666666666666666) * b) - pow(b, -1.0)) * (cos((a + b)) / -1.0));
}

Julia

function code(r, a, b)
	return Float64(r / Float64(Float64(Float64(fma(fma(-0.00205026455026455, Float64(b * b), -0.019444444444444445), Float64(b * b), -0.16666666666666666) * b) - (b ^ -1.0)) * Float64(cos(Float64(a + b)) / -1.0)))
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(N[(N[(N[(-0.00205026455026455 * N[(b * b), $MachinePrecision] + -0.019444444444444445), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * b), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 77.7

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

4. Applied rewrites 77.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 77.7

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

6. Applied rewrites 77.7%

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

7. Taylor expanded in b around 0

   \[\leadsto \frac{r}{\color{blue}{\frac{{b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{b}} \cdot \frac{\cos \left(a + b\right)}{-1}} \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{r}{\color{blue}{\left(\frac{{b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)}{b} - \frac{1}{b}\right)} \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{r}{\left(\frac{\color{blue}{\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot {b}^{2}}}{b} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{r}{\left(\color{blue}{\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \frac{{b}^{2}}{b}} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{r}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \frac{\color{blue}{1 \cdot {b}^{2}}}{b} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{r}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \color{blue}{\left(\frac{1}{b} \cdot {b}^{2}\right)} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   6. unpow2 N/A

      \[\leadsto \frac{r}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \left(\frac{1}{b} \cdot \color{blue}{\left(b \cdot b\right)}\right) - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   7. associate-*r* N/A

      \[\leadsto \frac{r}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \color{blue}{\left(\left(\frac{1}{b} \cdot b\right) \cdot b\right)} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   8. lft-mult-inverse N/A

      \[\leadsto \frac{r}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{1} \cdot b\right) - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   9. *-lft-identity N/A

      \[\leadsto \frac{r}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot \color{blue}{b} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{r}{\color{blue}{\left(\left({b}^{2} \cdot \left(\frac{-31}{15120} \cdot {b}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) \cdot b - \frac{1}{b}\right)} \cdot \frac{\cos \left(a + b\right)}{-1}} \]

9. Applied rewrites 52.4%

   \[\leadsto \frac{r}{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, b \cdot b, -0.019444444444444445\right), b \cdot b, -0.16666666666666666\right) \cdot b - \frac{1}{b}\right)} \cdot \frac{\cos \left(a + b\right)}{-1}} \]

10. Final simplification 52.4%

    \[\leadsto \frac{r}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, b \cdot b, -0.019444444444444445\right), b \cdot b, -0.16666666666666666\right) \cdot b - {b}^{-1}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]
11. Add Preprocessing

Alternative 3: 51.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\left(-0.16666666666666666 \cdot b - {b}^{-1}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (* (- (* -0.16666666666666666 b) (pow b -1.0)) (/ (cos (+ a b)) -1.0))))

C

double code(double r, double a, double b) {
	return r / (((-0.16666666666666666 * b) - pow(b, -1.0)) * (cos((a + b)) / -1.0));
}

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 / ((((-0.16666666666666666d0) * b) - (b ** (-1.0d0))) * (cos((a + b)) / (-1.0d0)))
end function

Java

public static double code(double r, double a, double b) {
	return r / (((-0.16666666666666666 * b) - Math.pow(b, -1.0)) * (Math.cos((a + b)) / -1.0));
}

Python

def code(r, a, b):
	return r / (((-0.16666666666666666 * b) - math.pow(b, -1.0)) * (math.cos((a + b)) / -1.0))

Julia

function code(r, a, b)
	return Float64(r / Float64(Float64(Float64(-0.16666666666666666 * b) - (b ^ -1.0)) * Float64(cos(Float64(a + b)) / -1.0)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r / (((-0.16666666666666666 * b) - (b ^ -1.0)) * (cos((a + b)) / -1.0));
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(N[(-0.16666666666666666 * b), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 77.7

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

4. Applied rewrites 77.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 77.7

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

6. Applied rewrites 77.7%

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

7. Taylor expanded in b around 0

   \[\leadsto \frac{r}{\color{blue}{\frac{\frac{-1}{6} \cdot {b}^{2} - 1}{b}} \cdot \frac{\cos \left(a + b\right)}{-1}} \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{r}{\color{blue}{\left(\frac{\frac{-1}{6} \cdot {b}^{2}}{b} - \frac{1}{b}\right)} \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{r}{\left(\color{blue}{\frac{-1}{6} \cdot \frac{{b}^{2}}{b}} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   3. *-lft-identity N/A

      \[\leadsto \frac{r}{\left(\frac{-1}{6} \cdot \frac{\color{blue}{1 \cdot {b}^{2}}}{b} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   4. associate-*l/ N/A

      \[\leadsto \frac{r}{\left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{b} \cdot {b}^{2}\right)} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   5. unpow2 N/A

      \[\leadsto \frac{r}{\left(\frac{-1}{6} \cdot \left(\frac{1}{b} \cdot \color{blue}{\left(b \cdot b\right)}\right) - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   6. associate-*r* N/A

      \[\leadsto \frac{r}{\left(\frac{-1}{6} \cdot \color{blue}{\left(\left(\frac{1}{b} \cdot b\right) \cdot b\right)} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   7. lft-mult-inverse N/A

      \[\leadsto \frac{r}{\left(\frac{-1}{6} \cdot \left(\color{blue}{1} \cdot b\right) - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{r}{\left(\frac{-1}{6} \cdot \color{blue}{b} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{r}{\color{blue}{\left(\frac{-1}{6} \cdot b - \frac{1}{b}\right)} \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{r}{\left(\color{blue}{\frac{-1}{6} \cdot b} - \frac{1}{b}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

   11. lower-/.f64 52.4

       \[\leadsto \frac{r}{\left(-0.16666666666666666 \cdot b - \color{blue}{\frac{1}{b}}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]

9. Applied rewrites 52.4%

   \[\leadsto \frac{r}{\color{blue}{\left(-0.16666666666666666 \cdot b - \frac{1}{b}\right)} \cdot \frac{\cos \left(a + b\right)}{-1}} \]

10. Final simplification 52.4%

    \[\leadsto \frac{r}{\left(-0.16666666666666666 \cdot b - {b}^{-1}\right) \cdot \frac{\cos \left(a + b\right)}{-1}} \]
11. Add Preprocessing

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -11000000000 \lor \neg \left(a \leq 0.0019\right):\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (or (<= a -11000000000.0) (not (<= a 0.0019)))
     (/ t_0 (cos a))
     (/ t_0 (cos b)))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if ((a <= -11000000000.0) || !(a <= 0.0019)) {
		tmp = t_0 / cos(a);
	} else {
		tmp = t_0 / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if ((a <= (-11000000000.0d0)) .or. (.not. (a <= 0.0019d0))) then
        tmp = t_0 / cos(a)
    else
        tmp = t_0 / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if ((a <= -11000000000.0) || !(a <= 0.0019)) {
		tmp = t_0 / Math.cos(a);
	} else {
		tmp = t_0 / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if (a <= -11000000000.0) or not (a <= 0.0019):
		tmp = t_0 / math.cos(a)
	else:
		tmp = t_0 / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if ((a <= -11000000000.0) || !(a <= 0.0019))
		tmp = Float64(t_0 / cos(a));
	else
		tmp = Float64(t_0 / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if ((a <= -11000000000.0) || ~((a <= 0.0019)))
		tmp = t_0 / cos(a);
	else
		tmp = t_0 / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -11000000000.0], N[Not[LessEqual[a, 0.0019]], $MachinePrecision]], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e10 or 0.0019 < a

   1. Initial program 58.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]
   4. Step-by-step derivation

      1. lower-cos.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -1.1e10 < a < 0.0019

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b}} \]
   4. Step-by-step derivation

      1. lower-cos.f64 97.3

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

   5. Applied rewrites 97.3%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11000000000 \lor \neg \left(a \leq 0.0019\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11000000000 \lor \neg \left(a \leq 0.0019\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -11000000000.0) (not (<= a 0.0019)))
   (/ (* r (sin b)) (cos a))
   (* (/ r (cos b)) (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -11000000000.0) || !(a <= 0.0019)) {
		tmp = (r * sin(b)) / cos(a);
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-11000000000.0d0)) .or. (.not. (a <= 0.0019d0))) then
        tmp = (r * sin(b)) / cos(a)
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -11000000000.0) || !(a <= 0.0019)) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -11000000000.0) or not (a <= 0.0019):
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -11000000000.0) || !(a <= 0.0019))
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -11000000000.0) || ~((a <= 0.0019)))
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -11000000000.0], N[Not[LessEqual[a, 0.0019]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e10 or 0.0019 < a

   1. Initial program 58.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]
   4. Step-by-step derivation

      1. lower-cos.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -1.1e10 < a < 0.0019

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 97.3

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

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\frac{r}{\cos b} \cdot \sin b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11000000000 \lor \neg \left(a \leq 0.0019\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-5} \lor \neg \left(b \leq 0.21\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.9e-5) (not (<= b 0.21)))
   (* (/ r (cos b)) (sin b))
   (* (/ b (cos a)) r)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.9e-5) || !(b <= 0.21)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (b / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.9d-5)) .or. (.not. (b <= 0.21d0))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (b / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.9e-5) || !(b <= 0.21)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (b / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.9e-5) or not (b <= 0.21):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (b / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.9e-5) || !(b <= 0.21))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(b / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.9e-5) || ~((b <= 0.21)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (b / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.9e-5], N[Not[LessEqual[b, 0.21]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9000000000000001e-5 or 0.209999999999999992 < b

   1. Initial program 56.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if -1.9000000000000001e-5 < b < 0.209999999999999992

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \frac{b}{\cos a} \cdot \color{blue}{r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-5} \lor \neg \left(b \leq 0.21\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 + b))) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.9

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

4. Applied rewrites 77.9%

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

Alternative 8: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.8

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

4. Applied rewrites 77.8%

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

Alternative 9: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{\cos a} \cdot r \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b / cos(a)) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 52.1

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

5. Applied rewrites 52.1%

   \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]
6. Step-by-step derivation

7. Applied rewrites 52.1%

   \[\leadsto \frac{b}{\cos a} \cdot \color{blue}{r} \]
8. Add Preprocessing

Alternative 10: 34.9% accurate, 36.7× speedup

Math

\[\begin{array}{l} \\ b \cdot r \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* b r))

C

double code(double r, double a, double b) {
	return b * r;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * r
end function

Java

public static double code(double r, double a, double b) {
	return b * r;
}

Python

def code(r, a, b):
	return b * r

Julia

function code(r, a, b)
	return Float64(b * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = b * r;
end

Wolfram

code[r_, a_, b_] := N[(b * r), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 52.1

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

5. Applied rewrites 52.1%

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

6. Taylor expanded in a around 0

   \[\leadsto b \cdot \color{blue}{r} \]
7. Step-by-step derivation

8. Applied rewrites 35.4%

   \[\leadsto b \cdot \color{blue}{r} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024302 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))