rsin A (should all be same)

Percentage Accurate: 77.3% → 99.5%

Time: 10.2s

Alternatives: 13

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: 77.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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) * cos(b)) / sin(b)) - sin(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.4

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

4. Applied rewrites 75.4%

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]
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. div-sub N/A

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

   9. lower--.f64 N/A

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

   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. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in a around inf

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

   1. lower-sin.f64 99.5

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

9. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{1}{\frac{\cos a}{\tan b} - \sin a} \end{array} \]

FPCore

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

C

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

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 * (1.0d0 / ((cos(a) / tan(b)) - sin(a)))
end function

Java

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

Python

def code(r, a, b):
	return r * (1.0 / ((math.cos(a) / math.tan(b)) - math.sin(a)))

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = r * (1.0 / ((cos(a) / tan(b)) - sin(a)));
end

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.4

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

4. Applied rewrites 75.4%

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]
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. div-sub N/A

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

   9. lower--.f64 N/A

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

   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. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in a around inf

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

   1. lower-sin.f64 99.5

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

9. Applied rewrites 99.5%

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

    1. lift-/.f64 N/A

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

    2. frac-2neg N/A

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

    3. div-inv N/A

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

    4. lower-*.f64 N/A

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

    5. lower-neg.f64 N/A

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

    6. frac-2neg N/A

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

    7. metadata-eval N/A

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

    8. remove-double-neg N/A

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

    9. lower-/.f64 99.4

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

    \[\leadsto r \cdot \frac{1}{\frac{\cos a}{\tan b} - \sin a} \]
13. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\frac{\cos a}{\tan b} - \sin a} \end{array} \]

FPCore

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

C

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

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) / tan(b)) - sin(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.4

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

4. Applied rewrites 75.4%

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]
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. div-sub N/A

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

   9. lower--.f64 N/A

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

   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. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in a around inf

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

   1. lower-sin.f64 99.5

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

9. Applied rewrites 99.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. clear-num N/A

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

    5. un-div-inv N/A

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

    6. lower-/.f64 N/A

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

    7. lift-sin.f64 N/A

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

    8. lift-cos.f64 N/A

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

    9. quot-tan N/A

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

    10. lower-tan.f64 99.5

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

11. Applied rewrites 99.5%

    \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{\tan b} - \sin a}} \]
12. Add Preprocessing

Alternative 5: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.36 \lor \neg \left(b \leq 0.055\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.36) (not (<= b 0.055)))
   (* (/ r (cos b)) (sin b))
   (/
    (*
     (*
      r
      (fma
       (fma 0.008333333333333333 (* b b) -0.16666666666666666)
       (* b b)
       1.0))
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.36) || !(b <= 0.055)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = ((r * fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0)) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.36) || !(b <= 0.055))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(Float64(r * fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0)) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.36], N[Not[LessEqual[b, 0.055]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[(r * N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.35999999999999999 or 0.0550000000000000003 < b

   1. Initial program 56.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 55.1

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

   5. Applied rewrites 55.1%

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

   if -0.35999999999999999 < b < 0.0550000000000000003

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

   5. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.36 \lor \neg \left(b \leq 0.055\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.36:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.055:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.36)
   (/ (* r (sin b)) (cos b))
   (if (<= b 0.055)
     (/
      (*
       (*
        r
        (fma
         (fma 0.008333333333333333 (* b b) -0.16666666666666666)
         (* b b)
         1.0))
       b)
      (cos (+ a b)))
     (* (/ r (cos b)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.36) {
		tmp = (r * sin(b)) / cos(b);
	} else if (b <= 0.055) {
		tmp = ((r * fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0)) * b) / cos((a + b));
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.36)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	elseif (b <= 0.055)
		tmp = Float64(Float64(Float64(r * fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0)) * b) / cos(Float64(a + b)));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.36], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.055], N[(N[(N[(r * N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.35999999999999999

   1. Initial program 53.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   if -0.35999999999999999 < b < 0.0550000000000000003

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

   5. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

   if 0.0550000000000000003 < b

   1. Initial program 58.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.2

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

   5. Applied rewrites 58.2%

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

Alternative 7: 77.3% 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]

Derivation

1. Initial program 75.5%

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

Alternative 8: 77.3% 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 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.5

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

4. Applied rewrites 75.5%

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

Alternative 9: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -22 \lor \neg \left(b \leq 4\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -22.0) (not (<= b 4.0)))
   (/ (* r (sin b)) 1.0)
   (/
    (*
     (*
      r
      (fma
       (fma 0.008333333333333333 (* b b) -0.16666666666666666)
       (* b b)
       1.0))
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -22.0) || !(b <= 4.0)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = ((r * fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0)) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -22.0) || !(b <= 4.0))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(Float64(r * fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0)) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -22.0], N[Not[LessEqual[b, 4.0]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(r * N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -22 or 4 < b

   1. Initial program 56.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.4%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot b, 0.16666666666666666, -0.5\right), a, -b\right), \color{blue}{a}, 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 12.6%

       \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -22 < b < 4

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

   5. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -22 \lor \neg \left(b \leq 4\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \lor \neg \left(b \leq 1.2\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.4) (not (<= b 1.2)))
   (/ (* r (sin b)) 1.0)
   (* (/ r (cos a)) b)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.4) || !(b <= 1.2)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (r / cos(a)) * 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 ((b <= (-1.4d0)) .or. (.not. (b <= 1.2d0))) then
        tmp = (r * sin(b)) / 1.0d0
    else
        tmp = (r / cos(a)) * b
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.4) || !(b <= 1.2)) {
		tmp = (r * Math.sin(b)) / 1.0;
	} else {
		tmp = (r / Math.cos(a)) * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.4) or not (b <= 1.2):
		tmp = (r * math.sin(b)) / 1.0
	else:
		tmp = (r / math.cos(a)) * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.4) || !(b <= 1.2))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(r / cos(a)) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.4) || ~((b <= 1.2)))
		tmp = (r * sin(b)) / 1.0;
	else
		tmp = (r / cos(a)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.4], N[Not[LessEqual[b, 1.2]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999 or 1.19999999999999996 < b

   1. Initial program 56.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.4%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot b, 0.16666666666666666, -0.5\right), a, -b\right), \color{blue}{a}, 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{r \cdot \sin b}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 12.6%

       \[\leadsto \frac{r \cdot \sin b}{1} \]

   if -1.3999999999999999 < b < 1.19999999999999996

   1. Initial program 98.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \lor \neg \left(b \leq 1.2\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 47.3

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

5. Applied rewrites 47.3%

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

Alternative 12: 51.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 * (b / cos(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 47.3

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

5. Applied rewrites 47.3%

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

7. Applied rewrites 47.3%

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

Alternative 13: 34.8% 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 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 47.3

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

5. Applied rewrites 47.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 31.3%

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

Reproduce

herbie shell --seed 2024303 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))