rsin B (should all be same)

Percentage Accurate: 76.3% → 99.4%

Time: 11.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. inv-pow N/A

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

   5. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\frac{1}{{\cos a}^{-1}}}\right)} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666 \cdot b, b\right)}{t\_0} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b)))
        (t_1 (/ (sin b) t_0))
        (t_2 (* (/ r (cos b)) (sin b))))
   (if (<= t_1 -0.02)
     t_2
     (if (<= t_1 2e-16)
       (* (/ (fma (* b b) (* -0.16666666666666666 b) b) t_0) r)
       t_2))))

C

double code(double r, double a, double b) {
	double t_0 = cos((a + b));
	double t_1 = sin(b) / t_0;
	double t_2 = (r / cos(b)) * sin(b);
	double tmp;
	if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 2e-16) {
		tmp = (fma((b * b), (-0.16666666666666666 * b), b) / t_0) * r;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = cos(Float64(a + b))
	t_1 = Float64(sin(b) / t_0)
	t_2 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 2e-16)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-0.16666666666666666 * b), b) / t_0) * r);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 2e-16], N[(N[(N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 * b), $MachinePrecision] + b), $MachinePrecision] / t$95$0), $MachinePrecision] * r), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0200000000000000004 or 2e-16 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 53.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 53.5

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

   5. Applied rewrites 53.5%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 2e-16

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, \frac{-1}{6}, b\right)}{\cos \left(a + b\right)} \]

      9. lower-pow.f64 99.6

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, -0.16666666666666666, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, -0.16666666666666666, b\right)}}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.02:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666 \cdot b, b\right)}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{r}{{\tan b}^{-1} - a}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666 \cdot b, b\right)}{t\_0} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b)))
        (t_1 (/ (sin b) t_0))
        (t_2 (/ r (- (pow (tan b) -1.0) a))))
   (if (<= t_1 -0.02)
     t_2
     (if (<= t_1 2e-16)
       (* (/ (fma (* b b) (* -0.16666666666666666 b) b) t_0) r)
       t_2))))

C

double code(double r, double a, double b) {
	double t_0 = cos((a + b));
	double t_1 = sin(b) / t_0;
	double t_2 = r / (pow(tan(b), -1.0) - a);
	double tmp;
	if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 2e-16) {
		tmp = (fma((b * b), (-0.16666666666666666 * b), b) / t_0) * r;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = cos(Float64(a + b))
	t_1 = Float64(sin(b) / t_0)
	t_2 = Float64(r / Float64((tan(b) ^ -1.0) - a))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 2e-16)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-0.16666666666666666 * b), b) / t_0) * r);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(r / N[(N[Power[N[Tan[b], $MachinePrecision], -1.0], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 2e-16], N[(N[(N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 * b), $MachinePrecision] + b), $MachinePrecision] / t$95$0), $MachinePrecision] * r), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0200000000000000004 or 2e-16 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 53.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 29.7

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

   4. Applied rewrites 29.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 50.7

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

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 50.6

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

   9. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{r}{{\tan b}^{-1} - a}} \]

   if -0.0200000000000000004 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 2e-16

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, \frac{-1}{6}, b\right)}{\cos \left(a + b\right)} \]

      9. lower-pow.f64 99.6

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, -0.16666666666666666, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, -0.16666666666666666, b\right)}}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.02:\\ \;\;\;\;\frac{r}{{\tan b}^{-1} - a}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666 \cdot b, b\right)}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{{\tan b}^{-1} - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 7: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.32e-5)
   (* (/ r (cos b)) (sin b))
   (if (<= b 7.5e-29) (* (/ b (cos a)) r) (/ (* (sin b) r) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.32e-5) {
		tmp = (r / cos(b)) * sin(b);
	} else if (b <= 7.5e-29) {
		tmp = (b / cos(a)) * r;
	} else {
		tmp = (sin(b) * r) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.32d-5)) then
        tmp = (r / cos(b)) * sin(b)
    else if (b <= 7.5d-29) then
        tmp = (b / cos(a)) * r
    else
        tmp = (sin(b) * r) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.32e-5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else if (b <= 7.5e-29) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.32e-5:
		tmp = (r / math.cos(b)) * math.sin(b)
	elif b <= 7.5e-29:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = (math.sin(b) * r) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.32e-5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	elseif (b <= 7.5e-29)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.32e-5)
		tmp = (r / cos(b)) * sin(b);
	elseif (b <= 7.5e-29)
		tmp = (b / cos(a)) * r;
	else
		tmp = (sin(b) * r) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.32e-5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-29], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.32000000000000007e-5

   1. Initial program 56.7%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 56.8

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

   5. Applied rewrites 56.8%

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

   if -1.32000000000000007e-5 < b < 7.50000000000000006e-29

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 7.50000000000000006e-29 < b

   1. Initial program 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 52.4

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

   7. Applied rewrites 52.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% 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 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 79.4

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

4. Applied rewrites 79.4%

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

Alternative 9: 76.3% 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 79.4%

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

3. Final simplification 79.4%

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

Alternative 10: 50.8% 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 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 79.4

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

4. Applied rewrites 79.4%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 58.3

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

7. Applied rewrites 58.3%

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

Alternative 11: 50.9% 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 79.4%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Final simplification 58.2%

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

Alternative 12: 50.8% 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 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
6. 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 58.2

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

7. Applied rewrites 58.2%

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

Alternative 13: 34.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{1} \cdot r \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return (b / 1.0) * 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 / 1.0d0) * r
end function

Java

public static double code(double r, double a, double b) {
	return (b / 1.0) * r;
}

Python

def code(r, a, b):
	return (b / 1.0) * r

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = (b / 1.0) * r;
end

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 40.2%

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

9. Final simplification 40.2%

   \[\leadsto \frac{b}{1} \cdot r \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))