rsin A (should all be same)

Percentage Accurate: 75.8% → 99.5%

Time: 11.3s

Alternatives: 14

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: 75.8% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.4

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

4. Applied rewrites 77.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower--.f64 99.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.4

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

4. Applied rewrites 77.4%

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]
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. lift-sin.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. lower-/.f64 N/A

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

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. 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. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. clear-num N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. quot-tan N/A

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

   14. lower-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-/l* N/A

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

   18. *-inverses N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5300:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 0.048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -5300.0)
   (/ (* (sin b) r) (cos b))
   (if (<= b 0.048)
     (/
      (*
       (fma
        (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
        (* b b)
        r)
       b)
      (cos (+ a b)))
     (* (/ (sin b) (cos b)) r))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -5300.0) {
		tmp = (sin(b) * r) / cos(b);
	} else if (b <= 0.048) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = (sin(b) / cos(b)) * r;
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -5300.0)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	elseif (b <= 0.048)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -5300.0], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.048], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5300

   1. Initial program 47.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 48.1

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

   5. Applied rewrites 48.1%

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

   if -5300 < b < 0.048000000000000001

   1. Initial program 97.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 97.6%

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

   if 0.048000000000000001 < b

   1. Initial program 62.2%

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

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

      5. cos-diff N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 61.0

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

   7. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 61.0

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

   9. Applied rewrites 61.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5300:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 0.048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos b} \cdot r\\ \mathbf{if}\;b \leq -5300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) (cos b)) r)))
   (if (<= b -5300.0)
     t_0
     (if (<= b 0.048)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / cos(b)) * r;
	double tmp;
	if (b <= -5300.0) {
		tmp = t_0;
	} else if (b <= 0.048) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / cos(b)) * r)
	tmp = 0.0
	if (b <= -5300.0)
		tmp = t_0;
	elseif (b <= 0.048)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -5300.0], t$95$0, If[LessEqual[b, 0.048], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5300 or 0.048000000000000001 < b

   1. Initial program 55.7%

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

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

      5. cos-diff N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 55.1

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

   7. Applied rewrites 55.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if -5300 < b < 0.048000000000000001

   1. Initial program 97.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 97.6%

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

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -5300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -5300.0)
     t_0
     (if (<= b 0.048)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -5300.0) {
		tmp = t_0;
	} else if (b <= 0.048) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -5300.0)
		tmp = t_0;
	elseif (b <= 0.048)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5300.0], t$95$0, If[LessEqual[b, 0.048], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5300 or 0.048000000000000001 < b

   1. Initial program 55.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 55.0

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

   5. Applied rewrites 55.0%

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

   if -5300 < b < 0.048000000000000001

   1. Initial program 97.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 97.6%

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

Alternative 6: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Final simplification 77.5%

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

Alternative 7: 75.8% 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 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. *-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 77.5

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

4. Applied rewrites 77.5%

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

Alternative 8: 75.8% 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 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.4

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

4. Applied rewrites 77.4%

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

Alternative 9: 50.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in b around 0

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

   1. lower-*.f64 52.5

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

5. Applied rewrites 52.5%

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

Alternative 10: 50.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 52.4

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

5. Applied rewrites 52.4%

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

7. Applied rewrites 52.4%

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

Alternative 11: 50.6% 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 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 52.4

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

5. Applied rewrites 52.4%

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

Alternative 12: 50.6% 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 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 52.4

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

5. Applied rewrites 52.4%

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

7. Applied rewrites 52.4%

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

8. Final simplification 52.4%

   \[\leadsto \frac{b}{\cos a} \cdot r \]
9. Add Preprocessing

Alternative 13: 34.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\frac{\mathsf{fma}\left(b \cdot b, -0.3333333333333333, 1\right)}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (/ (fma (* b b) -0.3333333333333333 1.0) b)))

C

double code(double r, double a, double b) {
	return r / (fma((b * b), -0.3333333333333333, 1.0) / b);
}

Julia

function code(r, a, b)
	return Float64(r / Float64(fma(Float64(b * b), -0.3333333333333333, 1.0) / b))
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(N[(b * b), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.4

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

4. Applied rewrites 77.4%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 53.2%

   \[\leadsto \frac{r}{\color{blue}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \cos a\right) \cdot b - \sin a, b, \cos a\right)}{b}}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{r}{\frac{1 + \frac{-1}{3} \cdot {b}^{2}}{b}} \]
9. Step-by-step derivation

10. Applied rewrites 33.2%

    \[\leadsto \frac{r}{\frac{\mathsf{fma}\left(b \cdot b, -0.3333333333333333, 1\right)}{b}} \]
11. Add Preprocessing

Alternative 14: 33.8% 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 77.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))