rsin B (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 11.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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 77.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 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 77.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -55:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{elif}\;b \leq 0.0225:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \left(-r\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -55.0)
   (* (/ (sin b) (cos b)) r)
   (if (<= b 0.0225)
     (/
      (*
       (*
        (fma
         (fma (* b b) -0.008333333333333333 0.16666666666666666)
         (* b b)
         -1.0)
        b)
       (- r))
      (cos (+ a b)))
     (* (/ r (cos b)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -55.0) {
		tmp = (sin(b) / cos(b)) * r;
	} else if (b <= 0.0225) {
		tmp = ((fma(fma((b * b), -0.008333333333333333, 0.16666666666666666), (b * b), -1.0) * b) * -r) / cos((a + b));
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -55.0)
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	elseif (b <= 0.0225)
		tmp = Float64(Float64(Float64(fma(fma(Float64(b * b), -0.008333333333333333, 0.16666666666666666), Float64(b * b), -1.0) * b) * Float64(-r)) / cos(Float64(a + b)));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -55.0], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[b, 0.0225], N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision] * b), $MachinePrecision] * (-r)), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -55

   1. Initial program 57.7%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 56.4

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

   5. Applied rewrites 56.4%

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

   if -55 < b < 0.022499999999999999

   1. Initial program 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 98.5%

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

   if 0.022499999999999999 < b

   1. Initial program 57.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -55:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{elif}\;b \leq 0.0225:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \left(-r\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0225:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \left(-r\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -55.0)
     t_0
     (if (<= b 0.0225)
       (/
        (*
         (*
          (fma
           (fma (* b b) -0.008333333333333333 0.16666666666666666)
           (* b b)
           -1.0)
          b)
         (- r))
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -55.0) {
		tmp = t_0;
	} else if (b <= 0.0225) {
		tmp = ((fma(fma((b * b), -0.008333333333333333, 0.16666666666666666), (b * b), -1.0) * b) * -r) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -55.0)
		tmp = t_0;
	elseif (b <= 0.0225)
		tmp = Float64(Float64(Float64(fma(fma(Float64(b * b), -0.008333333333333333, 0.16666666666666666), Float64(b * b), -1.0) * b) * Float64(-r)) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -55.0], t$95$0, If[LessEqual[b, 0.0225], N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision] * b), $MachinePrecision] * (-r)), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -55 or 0.022499999999999999 < b

   1. Initial program 57.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 57.3

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

   5. Applied rewrites 57.3%

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

   if -55 < b < 0.022499999999999999

   1. Initial program 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 98.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -55:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 0.0225:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \left(-r\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% 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 77.9%

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

3. Final simplification 77.9%

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

Alternative 7: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{if}\;b \leq -39000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 22:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \left(-r\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (- r) (- (sin b)))))
   (if (<= b -39000000.0)
     t_0
     (if (<= b 22.0)
       (/
        (*
         (*
          (fma
           (fma (* b b) -0.008333333333333333 0.16666666666666666)
           (* b b)
           -1.0)
          b)
         (- r))
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = -r * -sin(b);
	double tmp;
	if (b <= -39000000.0) {
		tmp = t_0;
	} else if (b <= 22.0) {
		tmp = ((fma(fma((b * b), -0.008333333333333333, 0.16666666666666666), (b * b), -1.0) * b) * -r) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(-r) * Float64(-sin(b)))
	tmp = 0.0
	if (b <= -39000000.0)
		tmp = t_0;
	elseif (b <= 22.0)
		tmp = Float64(Float64(Float64(fma(fma(Float64(b * b), -0.008333333333333333, 0.16666666666666666), Float64(b * b), -1.0) * b) * Float64(-r)) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[((-r) * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[b, -39000000.0], t$95$0, If[LessEqual[b, 22.0], N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision] * b), $MachinePrecision] * (-r)), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9e7 or 22 < b

   1. Initial program 58.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 58.4

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 10.8

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

   7. Applied rewrites 10.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

       \[\leadsto \left(-\sin b\right) \cdot \left(-r\right) \]

   if -3.9e7 < b < 22

   1. Initial program 96.2%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 95.1%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -39000000:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{elif}\;b \leq 22:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.008333333333333333, 0.16666666666666666\right), b \cdot b, -1\right) \cdot b\right) \cdot \left(-r\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{if}\;b \leq -29000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 450000:\\ \;\;\;\;\left(\frac{-1}{\cos \left(a + b\right)} \cdot r\right) \cdot \left(\mathsf{fma}\left(b \cdot b, 0.16666666666666666, -1\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (- r) (- (sin b)))))
   (if (<= b -29000000000000.0)
     t_0
     (if (<= b 450000.0)
       (*
        (* (/ -1.0 (cos (+ a b))) r)
        (* (fma (* b b) 0.16666666666666666 -1.0) b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = -r * -sin(b);
	double tmp;
	if (b <= -29000000000000.0) {
		tmp = t_0;
	} else if (b <= 450000.0) {
		tmp = ((-1.0 / cos((a + b))) * r) * (fma((b * b), 0.16666666666666666, -1.0) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(-r) * Float64(-sin(b)))
	tmp = 0.0
	if (b <= -29000000000000.0)
		tmp = t_0;
	elseif (b <= 450000.0)
		tmp = Float64(Float64(Float64(-1.0 / cos(Float64(a + b))) * r) * Float64(fma(Float64(b * b), 0.16666666666666666, -1.0) * b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[((-r) * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[b, -29000000000000.0], t$95$0, If[LessEqual[b, 450000.0], N[(N[(N[(-1.0 / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.9e13 or 4.5e5 < b

   1. Initial program 57.8%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 10.9

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

   7. Applied rewrites 10.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.5%

       \[\leadsto \left(-\sin b\right) \cdot \left(-r\right) \]

   if -2.9e13 < b < 4.5e5

   1. Initial program 96.3%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.6

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

   7. Applied rewrites 93.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -29000000000000:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{elif}\;b \leq 450000:\\ \;\;\;\;\left(\frac{-1}{\cos \left(a + b\right)} \cdot r\right) \cdot \left(\mathsf{fma}\left(b \cdot b, 0.16666666666666666, -1\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\cos a} \cdot \left(-b\right)\right) \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3.3e+14) (* (- r) (- (sin b))) (* (* (/ -1.0 (cos a)) (- b)) r)))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = -r * -sin(b);
	} else {
		tmp = ((-1.0 / cos(a)) * -b) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-3.3d+14)) then
        tmp = -r * -sin(b)
    else
        tmp = (((-1.0d0) / cos(a)) * -b) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = -r * -Math.sin(b);
	} else {
		tmp = ((-1.0 / Math.cos(a)) * -b) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3.3e+14:
		tmp = -r * -math.sin(b)
	else:
		tmp = ((-1.0 / math.cos(a)) * -b) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3.3e+14)
		tmp = Float64(Float64(-r) * Float64(-sin(b)));
	else
		tmp = Float64(Float64(Float64(-1.0 / cos(a)) * Float64(-b)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3.3e+14)
		tmp = -r * -sin(b);
	else
		tmp = ((-1.0 / cos(a)) * -b) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3.3e+14], N[((-r) * (-N[Sin[b], $MachinePrecision])), $MachinePrecision], N[(N[(N[(-1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3e14

   1. Initial program 56.8%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.9%

       \[\leadsto \left(-\sin b\right) \cdot \left(-r\right) \]

   if -3.3e14 < b

   1. Initial program 86.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\cos a} \cdot \left(-b\right)\right) \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3.3e+14) (* (- r) (- (sin b))) (* (/ b (cos a)) r)))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = -r * -sin(b);
	} else {
		tmp = (b / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-3.3d+14)) then
        tmp = -r * -sin(b)
    else
        tmp = (b / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = -r * -Math.sin(b);
	} else {
		tmp = (b / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3.3e+14:
		tmp = -r * -math.sin(b)
	else:
		tmp = (b / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3.3e+14)
		tmp = Float64(Float64(-r) * Float64(-sin(b)));
	else
		tmp = Float64(Float64(b / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3.3e+14)
		tmp = -r * -sin(b);
	else
		tmp = (b / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3.3e+14], N[((-r) * (-N[Sin[b], $MachinePrecision])), $MachinePrecision], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3e14

   1. Initial program 56.8%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.9%

       \[\leadsto \left(-\sin b\right) \cdot \left(-r\right) \]

   if -3.3e14 < b

   1. Initial program 86.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3.3e+14) (* (- r) (- (sin b))) (* (/ r (cos a)) b)))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = -r * -sin(b);
	} else {
		tmp = (r / cos(a)) * b;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-3.3d+14)) then
        tmp = -r * -sin(b)
    else
        tmp = (r / cos(a)) * b
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = -r * -Math.sin(b);
	} else {
		tmp = (r / Math.cos(a)) * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3.3e+14:
		tmp = -r * -math.sin(b)
	else:
		tmp = (r / math.cos(a)) * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3.3e+14)
		tmp = Float64(Float64(-r) * Float64(-sin(b)));
	else
		tmp = Float64(Float64(r / cos(a)) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3.3e+14)
		tmp = -r * -sin(b);
	else
		tmp = (r / cos(a)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3.3e+14], N[((-r) * (-N[Sin[b], $MachinePrecision])), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3e14

   1. Initial program 56.8%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.9%

       \[\leadsto \left(-\sin b\right) \cdot \left(-r\right) \]

   if -3.3e14 < b

   1. Initial program 86.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. inv-pow N/A

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

      13. lower-pow.f64 86.2

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

   4. Applied rewrites 86.2%

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

   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 69.0

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

   7. Applied rewrites 69.0%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\left(-r\right) \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-r\right) \cdot \left(-\sin b\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 77.9

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

4. Applied rewrites 77.9%

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

5. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 53.5

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

7. Applied rewrites 53.5%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 36.4%

    \[\leadsto \left(-\sin b\right) \cdot \left(-r\right) \]

11. Final simplification 36.4%

    \[\leadsto \left(-r\right) \cdot \left(-\sin b\right) \]
12. Add Preprocessing

Alternative 13: 34.9% 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 77.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 50.5

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.9%

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

9. Final simplification 32.9%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))