rsin B (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 11.9s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0175:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -0.035)
     t_0
     (if (<= b 0.0175)
       (/
        (*
         b
         (fma
          (* b b)
          (* r (fma b (* b 0.008333333333333333) -0.16666666666666666))
          r))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(b));
	double tmp;
	if (b <= -0.035) {
		tmp = t_0;
	} else if (b <= 0.0175) {
		tmp = (b * fma((b * b), (r * fma(b, (b * 0.008333333333333333), -0.16666666666666666)), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(b)))
	tmp = 0.0
	if (b <= -0.035)
		tmp = t_0;
	elseif (b <= 0.0175)
		tmp = Float64(Float64(b * fma(Float64(b * b), Float64(r * fma(b, Float64(b * 0.008333333333333333), -0.16666666666666666)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.035], t$95$0, If[LessEqual[b, 0.0175], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(r * N[(b * N[(b * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.035000000000000003 or 0.017500000000000002 < b

   1. Initial program 62.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   if -0.035000000000000003 < b < 0.017500000000000002

   1. Initial program 98.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. lower-*.f64 98.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

Alternative 3: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0175:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (cos b))))
   (if (<= b -0.035)
     t_0
     (if (<= b 0.0175)
       (/
        (*
         b
         (fma
          (* b b)
          (* r (fma b (* b 0.008333333333333333) -0.16666666666666666))
          r))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / cos(b);
	double tmp;
	if (b <= -0.035) {
		tmp = t_0;
	} else if (b <= 0.0175) {
		tmp = (b * fma((b * b), (r * fma(b, (b * 0.008333333333333333), -0.16666666666666666)), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / cos(b))
	tmp = 0.0
	if (b <= -0.035)
		tmp = t_0;
	elseif (b <= 0.0175)
		tmp = Float64(Float64(b * fma(Float64(b * b), Float64(r * fma(b, Float64(b * 0.008333333333333333), -0.16666666666666666)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.035], t$95$0, If[LessEqual[b, 0.0175], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(r * N[(b * N[(b * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.035000000000000003 or 0.017500000000000002 < b

   1. Initial program 62.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -0.035000000000000003 < b < 0.017500000000000002

   1. Initial program 98.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. lower-*.f64 98.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

Alternative 4: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Final simplification 77.7%

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

Alternative 5: 56.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -480000000.0)
     t_0
     (if (<= b 55000000000.0)
       (/
        (*
         r
         (fma
          (fma
           (* b b)
           (fma (* b b) -0.0001984126984126984 0.008333333333333333)
           -0.16666666666666666)
          (* b (* b b))
          b))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -480000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = (r * fma(fma((b * b), fma((b * b), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (b * (b * b)), b)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -480000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(Float64(r * fma(fma(Float64(b * b), fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(b * Float64(b * b)), b)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -480000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(N[(r * N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.8e8 or 5.5e10 < b

   1. Initial program 61.8%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -4.8e8 < b < 5.5e10

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. lower-*.f64 96.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot {b}^{\color{blue}{3}} + 1 \cdot b\right)}{\cos \left(b + a\right)} \]

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 92.1%

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

Alternative 6: 56.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -480000000.0)
     t_0
     (if (<= b 55000000000.0)
       (*
        r
        (/
         (fma
          (fma
           b
           (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
           -0.16666666666666666)
          (* b (* b b))
          b)
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -480000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = r * (fma(fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -480000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(r * Float64(fma(fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -480000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(r * N[(N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.8e8 or 5.5e10 < b

   1. Initial program 61.8%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -4.8e8 < b < 5.5e10

   1. Initial program 96.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -11000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -11000.0)
     t_0
     (if (<= b 1.2e+18)
       (/
        (*
         b
         (fma
          (* b b)
          (* r (fma b (* b 0.008333333333333333) -0.16666666666666666))
          r))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -11000.0) {
		tmp = t_0;
	} else if (b <= 1.2e+18) {
		tmp = (b * fma((b * b), (r * fma(b, (b * 0.008333333333333333), -0.16666666666666666)), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -11000.0)
		tmp = t_0;
	elseif (b <= 1.2e+18)
		tmp = Float64(Float64(b * fma(Float64(b * b), Float64(r * fma(b, Float64(b * 0.008333333333333333), -0.16666666666666666)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -11000.0], t$95$0, If[LessEqual[b, 1.2e+18], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(r * N[(b * N[(b * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -11000 or 1.2e18 < b

   1. Initial program 62.5%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -11000 < b < 1.2e18

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. lower-*.f64 95.2

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 91.9

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

   7. Applied rewrites 91.9%

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

Alternative 8: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -11000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -11000.0)
     t_0
     (if (<= b 1.2e+18)
       (*
        r
        (/
         (fma
          (fma (* b b) 0.008333333333333333 -0.16666666666666666)
          (* b (* b b))
          b)
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -11000.0) {
		tmp = t_0;
	} else if (b <= 1.2e+18) {
		tmp = r * (fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -11000.0)
		tmp = t_0;
	elseif (b <= 1.2e+18)
		tmp = Float64(r * Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -11000.0], t$95$0, If[LessEqual[b, 1.2e+18], N[(r * N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -11000 or 1.2e18 < b

   1. Initial program 62.5%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -11000 < b < 1.2e18

   1. Initial program 95.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -11000:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(r, \left(b \cdot b\right) \cdot -0.16666666666666666, r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -880000000.0)
     t_0
     (if (<= b 55000000000.0)
       (/ (* b (fma r (* (* b b) -0.16666666666666666) r)) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = (b * fma(r, ((b * b) * -0.16666666666666666), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(Float64(b * fma(r, Float64(Float64(b * b) * -0.16666666666666666), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(N[(b * N[(r * N[(N[(b * b), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 5.5e10 < b

   1. Initial program 61.8%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -8.8e8 < b < 5.5e10

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. lower-*.f64 96.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(r, \left(b \cdot b\right) \cdot -0.16666666666666666, r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -880000000.0)
     t_0
     (if (<= b 55000000000.0)
       (* r (/ (fma b (* b (* b -0.16666666666666666)) b) (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 5.5e10 < b

   1. Initial program 61.8%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -8.8e8 < b < 5.5e10

   1. Initial program 96.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+19}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -880000000.0) t_0 (if (<= b 1.05e+19) (/ (* r b) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 1.05e+19) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / 1.0d0)
    if (b <= (-880000000.0d0)) then
        tmp = t_0
    else if (b <= 1.05d+19) then
        tmp = (r * b) / cos(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 1.05e+19) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / 1.0)
	tmp = 0
	if b <= -880000000.0:
		tmp = t_0
	elif b <= 1.05e+19:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 1.05e+19)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / 1.0);
	tmp = 0.0;
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 1.05e+19)
		tmp = (r * b) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$0, If[LessEqual[b, 1.05e+19], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 1.05e19 < b

   1. Initial program 62.3%

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

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.5%

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

   if -8.8e8 < b < 1.05e19

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. lower-*.f64 95.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 12: 51.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. lower-*.f64 77.7

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 44.5

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

7. Applied rewrites 44.5%

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

Alternative 13: 51.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 44.5

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

5. Applied rewrites 44.5%

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

Alternative 14: 35.2% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 44.5

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

5. Applied rewrites 44.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.2%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))