rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 11.4s

Alternatives: 9

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.4

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 77.1

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

4. Applied egg-rr 77.1%

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

   1. +-commutative N/A

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

   2. cos-sum N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-neg.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. sin-neg N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-neg.f64 N/A

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

   19. lower-*.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 77.1

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

4. Applied egg-rr 77.1%

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

   1. +-commutative N/A

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

   2. cos-sum N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-fma.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -0.00245:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0012:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -0.00245) t_0 (if (<= a 0.0012) (* r (tan b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(a));
	double tmp;
	if (a <= -0.00245) {
		tmp = t_0;
	} else if (a <= 0.0012) {
		tmp = r * tan(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * (r / cos(a))
    if (a <= (-0.00245d0)) then
        tmp = t_0
    else if (a <= 0.0012d0) then
        tmp = r * tan(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * (r / Math.cos(a));
	double tmp;
	if (a <= -0.00245) {
		tmp = t_0;
	} else if (a <= 0.0012) {
		tmp = r * Math.tan(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(a))
	tmp = 0
	if a <= -0.00245:
		tmp = t_0
	elif a <= 0.0012:
		tmp = r * math.tan(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(a)))
	tmp = 0.0
	if (a <= -0.00245)
		tmp = t_0;
	elseif (a <= 0.0012)
		tmp = Float64(r * tan(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(a));
	tmp = 0.0;
	if (a <= -0.00245)
		tmp = t_0;
	elseif (a <= 0.0012)
		tmp = r * tan(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00245], t$95$0, If[LessEqual[a, 0.0012], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0024499999999999999 or 0.00119999999999999989 < a

   1. Initial program 53.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 53.5

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

   5. Simplified 53.5%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.5

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

   7. Applied egg-rr 53.5%

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

   if -0.0024499999999999999 < a < 0.00119999999999999989

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 98.0

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

   5. Simplified 98.0%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. lower-tan.f64 98.2

         \[\leadsto r \cdot \color{blue}{\tan b} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{r \cdot \tan b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 77.2

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

4. Applied egg-rr 77.2%

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

5. Final simplification 77.2%

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

Alternative 6: 75.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-20}:\\ \;\;\;\;b \cdot \left(r \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -6.5e-7) t_0 (if (<= b 8e-20) (* b (* r (/ 1.0 (cos a)))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -6.5e-7) {
		tmp = t_0;
	} else if (b <= 8e-20) {
		tmp = b * (r * (1.0 / cos(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-6.5d-7)) then
        tmp = t_0
    else if (b <= 8d-20) then
        tmp = b * (r * (1.0d0 / cos(a)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -6.5e-7) {
		tmp = t_0;
	} else if (b <= 8e-20) {
		tmp = b * (r * (1.0 / Math.cos(a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -6.5e-7:
		tmp = t_0
	elif b <= 8e-20:
		tmp = b * (r * (1.0 / math.cos(a)))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -6.5e-7)
		tmp = t_0;
	elseif (b <= 8e-20)
		tmp = Float64(b * Float64(r * Float64(1.0 / cos(a))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -6.5e-7)
		tmp = t_0;
	elseif (b <= 8e-20)
		tmp = b * (r * (1.0 / cos(a)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.5e-7], t$95$0, If[LessEqual[b, 8e-20], N[(b * N[(r * N[(1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.50000000000000024e-7 or 7.99999999999999956e-20 < b

   1. Initial program 57.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 56.8

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

   5. Simplified 56.8%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. lower-tan.f64 56.9

         \[\leadsto r \cdot \color{blue}{\tan b} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{r \cdot \tan b} \]

   if -6.50000000000000024e-7 < b < 7.99999999999999956e-20

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.7

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

   5. Simplified 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{b \cdot \left(\frac{1}{\cos a} \cdot r\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-20}:\\ \;\;\;\;b \cdot \left(r \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-20}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -6.5e-7) t_0 (if (<= b 8e-20) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -6.5e-7) {
		tmp = t_0;
	} else if (b <= 8e-20) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-6.5d-7)) then
        tmp = t_0
    else if (b <= 8d-20) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -6.5e-7) {
		tmp = t_0;
	} else if (b <= 8e-20) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -6.5e-7:
		tmp = t_0
	elif b <= 8e-20:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -6.5e-7)
		tmp = t_0;
	elseif (b <= 8e-20)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -6.5e-7)
		tmp = t_0;
	elseif (b <= 8e-20)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.5e-7], t$95$0, If[LessEqual[b, 8e-20], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.50000000000000024e-7 or 7.99999999999999956e-20 < b

   1. Initial program 57.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 56.8

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

   5. Simplified 56.8%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. lower-tan.f64 56.9

         \[\leadsto r \cdot \color{blue}{\tan b} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{r \cdot \tan b} \]

   if -6.50000000000000024e-7 < b < 7.99999999999999956e-20

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.7

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

   5. Simplified 99.7%

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

Alternative 8: 59.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 61.4

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

5. Simplified 61.4%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. quot-tan N/A

      \[\leadsto r \cdot \color{blue}{\tan b} \]

   9. lower-tan.f64 61.4

      \[\leadsto r \cdot \color{blue}{\tan b} \]

7. Applied egg-rr 61.4%

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

Alternative 9: 33.8% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 49.6

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

5. Simplified 49.6%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{r \cdot b} \]

   2. lower-*.f64 33.9

      \[\leadsto \color{blue}{r \cdot b} \]

8. Simplified 33.9%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))