rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 10.9s

Alternatives: 14

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: 76.8% 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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) / fma(cos(b), cos(a), Float64(sin(a) * Float64(-sin(b))))) * 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[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

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

   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%

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

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{\sin b}{\cos b} \cdot r\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b)))
        (t_1 (/ (sin b) t_0))
        (t_2 (* (/ (sin b) (cos b)) r)))
   (if (<= t_1 -0.01)
     t_2
     (if (<= t_1 5e-7)
       (* (* (fma (* b b) -0.16666666666666666 1.0) b) (/ r t_0))
       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 = (sin(b) / cos(b)) * r;
	double tmp;
	if (t_1 <= -0.01) {
		tmp = t_2;
	} else if (t_1 <= 5e-7) {
		tmp = (fma((b * b), -0.16666666666666666, 1.0) * b) * (r / t_0);
	} 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(sin(b) / cos(b)) * r)
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = t_2;
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) * Float64(r / t_0));
	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[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], t$95$2, If[LessEqual[t$95$1, 5e-7], N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(r / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0100000000000000002 or 4.99999999999999977e-7 < (/.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

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

      1. lower-cos.f64 52.8

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

   5. Applied rewrites 52.8%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.99999999999999977e-7

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.01:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -0.01)
     t_2
     (if (<= t_1 5e-7)
       (* (* (fma (* b b) -0.16666666666666666 1.0) b) (/ r t_0))
       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 <= -0.01) {
		tmp = t_2;
	} else if (t_1 <= 5e-7) {
		tmp = (fma((b * b), -0.16666666666666666, 1.0) * b) * (r / t_0);
	} 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 <= -0.01)
		tmp = t_2;
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) * Float64(r / t_0));
	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, -0.01], t$95$2, If[LessEqual[t$95$1, 5e-7], N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(r / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0100000000000000002 or 4.99999999999999977e-7 < (/.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 52.7

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

   5. Applied rewrites 52.7%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.99999999999999977e-7

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.01:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \frac{r}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b))) (t_1 (/ (sin b) t_0)) (t_2 (* (/ r 1.0) (sin b))))
   (if (<= t_1 -0.1)
     t_2
     (if (<= t_1 0.02)
       (*
        (*
         (fma
          (fma 0.008333333333333333 (* b b) -0.16666666666666666)
          (* b b)
          1.0)
         b)
        (/ r t_0))
       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 / 1.0) * sin(b);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.02) {
		tmp = (fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0) * b) * (r / t_0);
	} 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 / 1.0) * sin(b))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0) * b) * Float64(r / t_0));
	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 / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 0.02], N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(r / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.10000000000000001 or 0.0200000000000000004 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 52.3%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 52.2

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

   4. Applied rewrites 52.2%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.0200000000000000004

   1. Initial program 98.3%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.1:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b}{t\_0} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b))) (t_1 (/ (sin b) t_0)) (t_2 (* (/ r 1.0) (sin b))))
   (if (<= t_1 -0.1)
     t_2
     (if (<= t_1 0.02)
       (*
        (/
         (*
          (fma
           (fma 0.008333333333333333 (* b b) -0.16666666666666666)
           (* b b)
           1.0)
          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 / 1.0) * sin(b);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.02) {
		tmp = ((fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0) * 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 / 1.0) * sin(b))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0) * 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 / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 0.02], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $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))) < -0.10000000000000001 or 0.0200000000000000004 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 52.3%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 52.2

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

   4. Applied rewrites 52.2%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.0200000000000000004

   1. Initial program 98.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.1:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

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

   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. Taylor expanded in a around inf

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

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. mul-1-neg N/A

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

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

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

   11. sin-neg N/A

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

   12. mul-1-neg N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. mul-1-neg N/A

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

   16. sin-neg N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-sin.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 7: 76.8% 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 74.4%

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

3. Final simplification 74.4%

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

Alternative 8: 54.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 360:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r 1.0) (sin b))))
   (if (<= b -320000000.0)
     t_0
     (if (<= b 360.0)
       (*
        (*
         (fma
          (fma
           (fma -0.0001984126984126984 (* b b) 0.008333333333333333)
           (* b b)
           -0.16666666666666666)
          (* b b)
          1.0)
         b)
        (/ r (cos (+ a b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / 1.0) * sin(b);
	double tmp;
	if (b <= -320000000.0) {
		tmp = t_0;
	} else if (b <= 360.0) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (b * b), 0.008333333333333333), (b * b), -0.16666666666666666), (b * b), 1.0) * b) * (r / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / 1.0) * sin(b))
	tmp = 0.0
	if (b <= -320000000.0)
		tmp = t_0;
	elseif (b <= 360.0)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(b * b), 0.008333333333333333), Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0) * b) * Float64(r / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -320000000.0], t$95$0, If[LessEqual[b, 360.0], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(b * b), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e8 or 360 < b

   1. Initial program 52.2%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 52.1

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

   4. Applied rewrites 52.1%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -3.2e8 < b < 360

   1. Initial program 98.4%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \color{blue}{\left(\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \color{blue}{\left(\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot b\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\left(\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot {b}^{2}} + 1\right) \cdot b\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, {b}^{2}, 1\right)} \cdot b\right) \]

      6. sub-neg N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\color{blue}{{b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {b}^{2}, 1\right) \cdot b\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) \cdot {b}^{2} + \color{blue}{\frac{-1}{6}}, {b}^{2}, 1\right) \cdot b\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, {b}^{2}, \frac{-1}{6}\right)}, {b}^{2}, 1\right) \cdot b\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{5040} \cdot {b}^{2} + \frac{1}{120}}, {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {b}^{2}, \frac{1}{120}\right)}, {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \]

      12. unpow2 N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{b \cdot b}, \frac{1}{120}\right), {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{b \cdot b}, \frac{1}{120}\right), {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, b \cdot b, \frac{1}{120}\right), \color{blue}{b \cdot b}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, b \cdot b, \frac{1}{120}\right), \color{blue}{b \cdot b}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b\right) \]

      16. unpow2 N/A

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, b \cdot b, \frac{1}{120}\right), b \cdot b, \frac{-1}{6}\right), \color{blue}{b \cdot b}, 1\right) \cdot b\right) \]

      17. lower-*.f64 97.7

          \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), \color{blue}{b \cdot b}, 1\right) \cdot b\right) \]

   7. Applied rewrites 97.7%

      \[\leadsto \frac{r}{\cos \left(a + b\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;b \leq 360:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 360:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r 1.0) (sin b))))
   (if (<= b -320000000.0)
     t_0
     (if (<= b 360.0)
       (*
        (/
         (*
          (fma
           (fma
            (fma -0.0001984126984126984 (* b b) 0.008333333333333333)
            (* b b)
            -0.16666666666666666)
           (* b b)
           1.0)
          b)
         (cos (+ a b)))
        r)
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / 1.0) * sin(b);
	double tmp;
	if (b <= -320000000.0) {
		tmp = t_0;
	} else if (b <= 360.0) {
		tmp = ((fma(fma(fma(-0.0001984126984126984, (b * b), 0.008333333333333333), (b * b), -0.16666666666666666), (b * b), 1.0) * b) / cos((a + b))) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / 1.0) * sin(b))
	tmp = 0.0
	if (b <= -320000000.0)
		tmp = t_0;
	elseif (b <= 360.0)
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(b * b), 0.008333333333333333), Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0) * b) / cos(Float64(a + b))) * r);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -320000000.0], t$95$0, If[LessEqual[b, 360.0], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(b * b), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e8 or 360 < b

   1. Initial program 52.2%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 52.1

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

   4. Applied rewrites 52.1%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -3.2e8 < b < 360

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot b}}{\cos \left(a + b\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot b}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot {b}^{2}} + 1\right) \cdot b}{\cos \left(a + b\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, {b}^{2}, 1\right)} \cdot b}{\cos \left(a + b\right)} \]

      6. sub-neg N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      7. *-commutative N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      8. metadata-eval N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) \cdot {b}^{2} + \color{blue}{\frac{-1}{6}}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, {b}^{2}, \frac{-1}{6}\right)}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      10. +-commutative N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{5040} \cdot {b}^{2} + \frac{1}{120}}, {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {b}^{2}, \frac{1}{120}\right)}, {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      12. unpow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{b \cdot b}, \frac{1}{120}\right), {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{b \cdot b}, \frac{1}{120}\right), {b}^{2}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      14. unpow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, b \cdot b, \frac{1}{120}\right), \color{blue}{b \cdot b}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, b \cdot b, \frac{1}{120}\right), \color{blue}{b \cdot b}, \frac{-1}{6}\right), {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      16. unpow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, b \cdot b, \frac{1}{120}\right), b \cdot b, \frac{-1}{6}\right), \color{blue}{b \cdot b}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      17. lower-*.f64 97.6

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), \color{blue}{b \cdot b}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

   5. Applied rewrites 97.6%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;b \leq 360:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right), b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 380:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r 1.0) (sin b))))
   (if (<= b -320000000.0)
     t_0
     (if (<= b 380.0)
       (* (* (fma (* b b) -0.16666666666666666 1.0) b) (/ r (cos (+ a b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / 1.0) * sin(b);
	double tmp;
	if (b <= -320000000.0) {
		tmp = t_0;
	} else if (b <= 380.0) {
		tmp = (fma((b * b), -0.16666666666666666, 1.0) * b) * (r / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / 1.0) * sin(b))
	tmp = 0.0
	if (b <= -320000000.0)
		tmp = t_0;
	elseif (b <= 380.0)
		tmp = Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) * Float64(r / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -320000000.0], t$95$0, If[LessEqual[b, 380.0], N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e8 or 380 < b

   1. Initial program 52.2%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 52.1

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

   4. Applied rewrites 52.1%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -3.2e8 < b < 380

   1. Initial program 98.4%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;b \leq 380:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot \frac{r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 380:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r 1.0) (sin b))))
   (if (<= b -320000000.0)
     t_0
     (if (<= b 380.0)
       (* (/ (* (fma (* b b) -0.16666666666666666 1.0) b) (cos (+ a b))) r)
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / 1.0) * sin(b);
	double tmp;
	if (b <= -320000000.0) {
		tmp = t_0;
	} else if (b <= 380.0) {
		tmp = ((fma((b * b), -0.16666666666666666, 1.0) * b) / cos((a + b))) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / 1.0) * sin(b))
	tmp = 0.0
	if (b <= -320000000.0)
		tmp = t_0;
	elseif (b <= 380.0)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) / cos(Float64(a + b))) * r);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -320000000.0], t$95$0, If[LessEqual[b, 380.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e8 or 380 < b

   1. Initial program 52.2%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 52.1

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

   4. Applied rewrites 52.1%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -3.2e8 < b < 380

   1. Initial program 98.4%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -320000000:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;b \leq 380:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;b \leq -4.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1500:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r 1.0) (sin b))))
   (if (<= b -4.6) t_0 (if (<= b 1500.0) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / 1.0) * sin(b);
	double tmp;
	if (b <= -4.6) {
		tmp = t_0;
	} else if (b <= 1500.0) {
		tmp = (r / cos(a)) * 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 = (r / 1.0d0) * sin(b)
    if (b <= (-4.6d0)) then
        tmp = t_0
    else if (b <= 1500.0d0) then
        tmp = (r / cos(a)) * 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 = (r / 1.0) * Math.sin(b);
	double tmp;
	if (b <= -4.6) {
		tmp = t_0;
	} else if (b <= 1500.0) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / 1.0) * math.sin(b)
	tmp = 0
	if b <= -4.6:
		tmp = t_0
	elif b <= 1500.0:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / 1.0) * sin(b))
	tmp = 0.0
	if (b <= -4.6)
		tmp = t_0;
	elseif (b <= 1500.0)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / 1.0) * sin(b);
	tmp = 0.0;
	if (b <= -4.6)
		tmp = t_0;
	elseif (b <= 1500.0)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.6], t$95$0, If[LessEqual[b, 1500.0], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5999999999999996 or 1500 < b

   1. Initial program 51.9%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 51.9

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

   4. Applied rewrites 51.9%

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

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

      4. cos-sum N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unsub-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 51.7

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.1%

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

   if -4.5999999999999996 < b < 1500

   1. Initial program 99.1%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   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 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 13: 50.5% 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 74.4%

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

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 74.4

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

4. Applied rewrites 74.4%

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

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 48.3

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

7. Applied rewrites 48.3%

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

Alternative 14: 35.0% 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 74.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 48.3

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

5. Applied rewrites 48.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 33.6%

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

9. Final simplification 33.6%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))