rsin B (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 11.4s

Alternatives: 15

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: 77.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.071:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot r\right) \cdot b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -1e-7)
     t_2
     (if (<= t_1 0.071)
       (/ (* (* (fma (* b b) -0.16666666666666666 1.0) r) b) 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 <= -1e-7) {
		tmp = t_2;
	} else if (t_1 <= 0.071) {
		tmp = ((fma((b * b), -0.16666666666666666, 1.0) * r) * b) / 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 <= -1e-7)
		tmp = t_2;
	elseif (t_1 <= 0.071)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * r) * b) / 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, -1e-7], t$95$2, If[LessEqual[t$95$1, 0.071], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * r), $MachinePrecision] * b), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -9.9999999999999995e-8 or 0.0709999999999999937 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 55.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 54.8

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

   5. Applied rewrites 54.8%

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

   if -9.9999999999999995e-8 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.0709999999999999937

   1. Initial program 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.4

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

   7. Applied rewrites 97.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin a\\ t_1 := \frac{r}{\frac{\mathsf{fma}\left(t\_0, b, \cos a\right)}{b}}\\ \mathbf{if}\;a \leq -0.48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\sin b, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right), a \cdot a, -0.5\right), a \cdot a, 1\right) \cdot \cos b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin a))) (t_1 (/ r (/ (fma t_0 b (cos a)) b))))
   (if (<= a -0.48)
     t_1
     (if (<= a 0.42)
       (*
        (/
         (sin b)
         (fma
          (sin b)
          t_0
          (*
           (fma
            (fma
             (fma -0.001388888888888889 (* a a) 0.041666666666666664)
             (* a a)
             -0.5)
            (* a a)
            1.0)
           (cos b))))
        r)
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = -sin(a);
	double t_1 = r / (fma(t_0, b, cos(a)) / b);
	double tmp;
	if (a <= -0.48) {
		tmp = t_1;
	} else if (a <= 0.42) {
		tmp = (sin(b) / fma(sin(b), t_0, (fma(fma(fma(-0.001388888888888889, (a * a), 0.041666666666666664), (a * a), -0.5), (a * a), 1.0) * cos(b)))) * r;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(-sin(a))
	t_1 = Float64(r / Float64(fma(t_0, b, cos(a)) / b))
	tmp = 0.0
	if (a <= -0.48)
		tmp = t_1;
	elseif (a <= 0.42)
		tmp = Float64(Float64(sin(b) / fma(sin(b), t_0, Float64(fma(fma(fma(-0.001388888888888889, Float64(a * a), 0.041666666666666664), Float64(a * a), -0.5), Float64(a * a), 1.0) * cos(b)))) * r);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[Sin[a], $MachinePrecision])}, Block[{t$95$1 = N[(r / N[(N[(t$95$0 * b + N[Cos[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.48], t$95$1, If[LessEqual[a, 0.42], N[(N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] * t$95$0 + N[(N[(N[(N[(-0.001388888888888889 * N[(a * a), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(a * a), $MachinePrecision] + -0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.47999999999999998 or 0.419999999999999984 < a

   1. Initial program 55.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.5

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -0.47999999999999998 < a < 0.419999999999999984

   1. Initial program 97.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\left(1 + {a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\left({a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \left(\color{blue}{\left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}\right) \cdot {a}^{2}} + 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\mathsf{fma}\left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}, {a}^{2}, 1\right)}\right)} \]

      4. sub-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\color{blue}{{a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {a}^{2}, 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) \cdot {a}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {a}^{2}, 1\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) \cdot {a}^{2} + \color{blue}{\frac{-1}{2}}, {a}^{2}, 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}, {a}^{2}, \frac{-1}{2}\right)}, {a}^{2}, 1\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {a}^{2} + \frac{1}{24}}, {a}^{2}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {a}^{2}, \frac{1}{24}\right)}, {a}^{2}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{a \cdot a}, \frac{1}{24}\right), {a}^{2}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{a \cdot a}, \frac{1}{24}\right), {a}^{2}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right), \color{blue}{a \cdot a}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right), \color{blue}{a \cdot a}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      14. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right), a \cdot a, \frac{-1}{2}\right), \color{blue}{a \cdot a}, 1\right)\right)} \]

      15. lower-*.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right), a \cdot a, -0.5\right), \color{blue}{a \cdot a}, 1\right)\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right), a \cdot a, -0.5\right), a \cdot a, 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.48:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right), a \cdot a, -0.5\right), a \cdot a, 1\right) \cdot \cos b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin a\\ t_1 := \frac{r}{\frac{\mathsf{fma}\left(t\_0, b, \cos a\right)}{b}}\\ \mathbf{if}\;a \leq -0.24:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.28:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\sin b, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, a \cdot a, -0.5\right), a \cdot a, 1\right) \cdot \cos b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin a))) (t_1 (/ r (/ (fma t_0 b (cos a)) b))))
   (if (<= a -0.24)
     t_1
     (if (<= a 0.28)
       (*
        (/
         (sin b)
         (fma
          (sin b)
          t_0
          (*
           (fma (fma 0.041666666666666664 (* a a) -0.5) (* a a) 1.0)
           (cos b))))
        r)
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = -sin(a);
	double t_1 = r / (fma(t_0, b, cos(a)) / b);
	double tmp;
	if (a <= -0.24) {
		tmp = t_1;
	} else if (a <= 0.28) {
		tmp = (sin(b) / fma(sin(b), t_0, (fma(fma(0.041666666666666664, (a * a), -0.5), (a * a), 1.0) * cos(b)))) * r;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(-sin(a))
	t_1 = Float64(r / Float64(fma(t_0, b, cos(a)) / b))
	tmp = 0.0
	if (a <= -0.24)
		tmp = t_1;
	elseif (a <= 0.28)
		tmp = Float64(Float64(sin(b) / fma(sin(b), t_0, Float64(fma(fma(0.041666666666666664, Float64(a * a), -0.5), Float64(a * a), 1.0) * cos(b)))) * r);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[Sin[a], $MachinePrecision])}, Block[{t$95$1 = N[(r / N[(N[(t$95$0 * b + N[Cos[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.24], t$95$1, If[LessEqual[a, 0.28], N[(N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] * t$95$0 + N[(N[(N[(0.041666666666666664 * N[(a * a), $MachinePrecision] + -0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.23999999999999999 or 0.28000000000000003 < a

   1. Initial program 55.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.5

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -0.23999999999999999 < a < 0.28000000000000003

   1. Initial program 97.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}\right) + 1\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}\right) \cdot {a}^{2}} + 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right)}\right)} \]

      4. sub-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {a}^{2}, 1\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {a}^{2} + \color{blue}{\frac{-1}{2}}, {a}^{2}, 1\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {a}^{2}, \frac{-1}{2}\right)}, {a}^{2}, 1\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{a \cdot a}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{a \cdot a}, \frac{-1}{2}\right), {a}^{2}, 1\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, a \cdot a, \frac{-1}{2}\right), \color{blue}{a \cdot a}, 1\right)\right)} \]

      10. lower-*.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, a \cdot a, -0.5\right), \color{blue}{a \cdot a}, 1\right)\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, a \cdot a, -0.5\right), a \cdot a, 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.24:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \mathbf{elif}\;a \leq 0.28:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, a \cdot a, -0.5\right), a \cdot a, 1\right) \cdot \cos b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \mathbf{if}\;a \leq -3.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.6:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(0.16666666666666666 \cdot a, a, -1\right) \cdot \sin b\right) \cdot a\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ (fma (- (sin a)) b (cos a)) b))))
   (if (<= a -3.3)
     t_0
     (if (<= a 3.6)
       (*
        (/
         (sin b)
         (fma
          (cos b)
          (cos a)
          (* (* (fma (* 0.16666666666666666 a) a -1.0) (sin b)) a)))
        r)
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / (fma(-sin(a), b, cos(a)) / b);
	double tmp;
	if (a <= -3.3) {
		tmp = t_0;
	} else if (a <= 3.6) {
		tmp = (sin(b) / fma(cos(b), cos(a), ((fma((0.16666666666666666 * a), a, -1.0) * sin(b)) * a))) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(fma(Float64(-sin(a)), b, cos(a)) / b))
	tmp = 0.0
	if (a <= -3.3)
		tmp = t_0;
	elseif (a <= 3.6)
		tmp = Float64(Float64(sin(b) / fma(cos(b), cos(a), Float64(Float64(fma(Float64(0.16666666666666666 * a), a, -1.0) * sin(b)) * a))) * r);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / N[(N[((-N[Sin[a], $MachinePrecision]) * b + N[Cos[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.3], t$95$0, If[LessEqual[a, 3.6], N[(N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[(N[(N[(0.16666666666666666 * a), $MachinePrecision] * a + -1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.2999999999999998 or 3.60000000000000009 < a

   1. Initial program 55.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.5

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -3.2999999999999998 < a < 3.60000000000000009

   1. Initial program 97.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \mathbf{elif}\;a \leq 3.6:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{fma}\left(0.16666666666666666 \cdot a, a, -1\right) \cdot \sin b\right) \cdot a\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin a\\ t_1 := \frac{r}{\frac{\mathsf{fma}\left(t\_0, b, \cos a\right)}{b}}\\ \mathbf{if}\;a \leq -0.00088:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.11:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\sin b, t\_0, \mathsf{fma}\left(a \cdot a, -0.5, 1\right) \cdot \cos b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin a))) (t_1 (/ r (/ (fma t_0 b (cos a)) b))))
   (if (<= a -0.00088)
     t_1
     (if (<= a 0.11)
       (* (/ (sin b) (fma (sin b) t_0 (* (fma (* a a) -0.5 1.0) (cos b)))) r)
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = -sin(a);
	double t_1 = r / (fma(t_0, b, cos(a)) / b);
	double tmp;
	if (a <= -0.00088) {
		tmp = t_1;
	} else if (a <= 0.11) {
		tmp = (sin(b) / fma(sin(b), t_0, (fma((a * a), -0.5, 1.0) * cos(b)))) * r;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(-sin(a))
	t_1 = Float64(r / Float64(fma(t_0, b, cos(a)) / b))
	tmp = 0.0
	if (a <= -0.00088)
		tmp = t_1;
	elseif (a <= 0.11)
		tmp = Float64(Float64(sin(b) / fma(sin(b), t_0, Float64(fma(Float64(a * a), -0.5, 1.0) * cos(b)))) * r);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[Sin[a], $MachinePrecision])}, Block[{t$95$1 = N[(r / N[(N[(t$95$0 * b + N[Cos[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00088], t$95$1, If[LessEqual[a, 0.11], N[(N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] * t$95$0 + N[(N[(N[(a * a), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.80000000000000031e-4 or 0.110000000000000001 < a

   1. Initial program 56.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 55.3

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

   5. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 55.3

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

   7. Applied rewrites 55.3%

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

   8. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 61.5

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

   10. Applied rewrites 61.5%

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

   if -8.80000000000000031e-4 < a < 0.110000000000000001

   1. Initial program 97.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \color{blue}{\mathsf{fma}\left(a \cdot a, -0.5, 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00088:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \mathbf{elif}\;a \leq 0.11:\\ \;\;\;\;\frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(a \cdot a, -0.5, 1\right) \cdot \cos b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 34000000000:\\ \;\;\;\;\frac{\sin b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a 34000000000.0)
   (* (/ (sin b) (cos (+ a b))) r)
   (/ r (/ (fma (- (sin a)) b (cos a)) b))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= 34000000000.0) {
		tmp = (sin(b) / cos((a + b))) * r;
	} else {
		tmp = r / (fma(-sin(a), b, cos(a)) / b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= 34000000000.0)
		tmp = Float64(Float64(sin(b) / cos(Float64(a + b))) * r);
	else
		tmp = Float64(r / Float64(fma(Float64(-sin(a)), b, cos(a)) / b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, 34000000000.0], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(r / N[(N[((-N[Sin[a], $MachinePrecision]) * b + N[Cos[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.4e10

   1. Initial program 83.7%

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

   if 3.4e10 < a

   1. Initial program 62.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 62.5

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

   5. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 62.4

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

   7. Applied rewrites 62.4%

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

   8. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 68.0

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

   10. Applied rewrites 68.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 34000000000:\\ \;\;\;\;\frac{\sin b}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-\sin a, b, \cos a\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos a} \cdot r\\ \mathbf{if}\;a \leq -0.00041:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0085:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) (cos a)) r)))
   (if (<= a -0.00041) t_0 (if (<= a 0.0085) (* (/ (sin b) (cos b)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / cos(a)) * r;
	double tmp;
	if (a <= -0.00041) {
		tmp = t_0;
	} else if (a <= 0.0085) {
		tmp = (sin(b) / cos(b)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / cos(a)) * r
    if (a <= (-0.00041d0)) then
        tmp = t_0
    else if (a <= 0.0085d0) then
        tmp = (sin(b) / cos(b)) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / Math.cos(a)) * r;
	double tmp;
	if (a <= -0.00041) {
		tmp = t_0;
	} else if (a <= 0.0085) {
		tmp = (Math.sin(b) / Math.cos(b)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / math.cos(a)) * r
	tmp = 0
	if a <= -0.00041:
		tmp = t_0
	elif a <= 0.0085:
		tmp = (math.sin(b) / math.cos(b)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / cos(a)) * r)
	tmp = 0.0
	if (a <= -0.00041)
		tmp = t_0;
	elseif (a <= 0.0085)
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / cos(a)) * r;
	tmp = 0.0;
	if (a <= -0.00041)
		tmp = t_0;
	elseif (a <= 0.0085)
		tmp = (sin(b) / cos(b)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -0.00041], t$95$0, If[LessEqual[a, 0.0085], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.0999999999999999e-4 or 0.0085000000000000006 < a

   1. Initial program 56.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if -4.0999999999999999e-4 < a < 0.0085000000000000006

   1. Initial program 97.7%

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

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

   5. Applied rewrites 97.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00041:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 0.0085:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos a} \cdot r\\ \mathbf{if}\;a \leq -0.00041:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0085:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) (cos a)) r)))
   (if (<= a -0.00041) t_0 (if (<= a 0.0085) (* (/ r (cos b)) (sin b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / cos(a)) * r;
	double tmp;
	if (a <= -0.00041) {
		tmp = t_0;
	} else if (a <= 0.0085) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / cos(a)) * r
    if (a <= (-0.00041d0)) then
        tmp = t_0
    else if (a <= 0.0085d0) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / Math.cos(a)) * r;
	double tmp;
	if (a <= -0.00041) {
		tmp = t_0;
	} else if (a <= 0.0085) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / math.cos(a)) * r
	tmp = 0
	if a <= -0.00041:
		tmp = t_0
	elif a <= 0.0085:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / cos(a)) * r)
	tmp = 0.0
	if (a <= -0.00041)
		tmp = t_0;
	elseif (a <= 0.0085)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / cos(a)) * r;
	tmp = 0.0;
	if (a <= -0.00041)
		tmp = t_0;
	elseif (a <= 0.0085)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -0.00041], t$95$0, If[LessEqual[a, 0.0085], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.0999999999999999e-4 or 0.0085000000000000006 < a

   1. Initial program 56.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if -4.0999999999999999e-4 < a < 0.0085000000000000006

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00041:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 0.0085:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Final simplification 77.4%

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

Alternative 13: 51.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 53.2

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

5. Applied rewrites 53.2%

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

6. Final simplification 53.2%

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

Alternative 14: 50.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 53.2

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

7. Applied rewrites 53.2%

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

Alternative 15: 34.7% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b / 1.0d0) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 53.2

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

5. Applied rewrites 53.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.2%

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

9. Final simplification 36.2%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))