rsin B (should all be same)

Percentage Accurate: 76.0% → 99.5%

Time: 10.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))))
   (if (or (<= t_0 -2e-8) (not (<= t_0 2e-6)))
     (* (/ r (cos b)) (sin b))
     (* r (/ b (cos a))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double tmp;
	if ((t_0 <= -2e-8) || !(t_0 <= 2e-6)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    if ((t_0 <= (-2d-8)) .or. (.not. (t_0 <= 2d-6))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) / Math.cos((a + b));
	double tmp;
	if ((t_0 <= -2e-8) || !(t_0 <= 2e-6)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	tmp = 0
	if (t_0 <= -2e-8) or not (t_0 <= 2e-6):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	tmp = 0.0
	if ((t_0 <= -2e-8) || !(t_0 <= 2e-6))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((a + b));
	tmp = 0.0;
	if ((t_0 <= -2e-8) || ~((t_0 <= 2e-6)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-8], N[Not[LessEqual[t$95$0, 2e-6]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -2e-8 or 1.99999999999999991e-6 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 60.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if -2e-8 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 1.99999999999999991e-6

   1. Initial program 99.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -2 \cdot 10^{-8} \lor \neg \left(\frac{\sin b}{\cos \left(a + b\right)} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 4: 75.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. associate-*r* N/A

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

6. Applied rewrites 80.0%

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

Alternative 5: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-5} \lor \neg \left(a \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.9e-5) (not (<= a 5.3e-5)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-5) || !(a <= 5.3e-5)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(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 ((a <= (-1.9d-5)) .or. (.not. (a <= 5.3d-5))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-5) || !(a <= 5.3e-5)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.9e-5) or not (a <= 5.3e-5):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.9e-5) || !(a <= 5.3e-5))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -1.9e-5) || ~((a <= 5.3e-5)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.9e-5], N[Not[LessEqual[a, 5.3e-5]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000001e-5 or 5.3000000000000001e-5 < a

   1. Initial program 63.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if -1.9000000000000001e-5 < a < 5.3000000000000001e-5

   1. Initial program 98.7%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-5} \lor \neg \left(a \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-5} \lor \neg \left(a \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.9e-5) (not (<= a 5.3e-5)))
   (* r (/ (sin b) (cos a)))
   (* (/ r (cos b)) (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-5) || !(a <= 5.3e-5)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = (r / cos(b)) * sin(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 ((a <= (-1.9d-5)) .or. (.not. (a <= 5.3d-5))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-5) || !(a <= 5.3e-5)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.9e-5) or not (a <= 5.3e-5):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.9e-5) || !(a <= 5.3e-5))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -1.9e-5) || ~((a <= 5.3e-5)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.9e-5], N[Not[LessEqual[a, 5.3e-5]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000001e-5 or 5.3000000000000001e-5 < a

   1. Initial program 63.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if -1.9000000000000001e-5 < a < 5.3000000000000001e-5

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-5} \lor \neg \left(a \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

Alternative 8: 55.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 17:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.85)
   (* (* -1.0 r) (sin b))
   (if (<= b 17.0)
     (/
      (/ r (cos (+ a b)))
      (/
       (fma
        (fma
         (fma 0.00205026455026455 (* b b) 0.019444444444444445)
         (* b b)
         0.16666666666666666)
        (* b b)
        1.0)
       b))
     (* (- r) (* -1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.85) {
		tmp = (-1.0 * r) * sin(b);
	} else if (b <= 17.0) {
		tmp = (r / cos((a + b))) / (fma(fma(fma(0.00205026455026455, (b * b), 0.019444444444444445), (b * b), 0.16666666666666666), (b * b), 1.0) / b);
	} else {
		tmp = -r * (-1.0 * sin(b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.85)
		tmp = Float64(Float64(-1.0 * r) * sin(b));
	elseif (b <= 17.0)
		tmp = Float64(Float64(r / cos(Float64(a + b))) / Float64(fma(fma(fma(0.00205026455026455, Float64(b * b), 0.019444444444444445), Float64(b * b), 0.16666666666666666), Float64(b * b), 1.0) / b));
	else
		tmp = Float64(Float64(-r) * Float64(-1.0 * sin(b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.85], N[(N[(-1.0 * r), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 17.0], N[(N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.00205026455026455 * N[(b * b), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8500000000000001

   1. Initial program 57.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-div N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-neg.f64 N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-pow N/A

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

      19. pow2 N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqr-neg N/A

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

   9. Applied rewrites 12.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]
   11. Step-by-step derivation

   12. Applied rewrites 13.8%

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]

   if -1.8500000000000001 < b < 17

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{1 + {b}^{2} \cdot \left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right)\right)}{b}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{1 + {b}^{2} \cdot \left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right)\right)}{b}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{{b}^{2} \cdot \left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right)\right) + 1}}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{\left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right)\right) \cdot {b}^{2}} + 1}{b}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {b}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right), {b}^{2}, 1\right)}}{b}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\color{blue}{{b}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right) + \frac{1}{6}}, {b}^{2}, 1\right)}{b}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}\right) \cdot {b}^{2}} + \frac{1}{6}, {b}^{2}, 1\right)}{b}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360} + \frac{31}{15120} \cdot {b}^{2}, {b}^{2}, \frac{1}{6}\right)}, {b}^{2}, 1\right)}{b}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{31}{15120} \cdot {b}^{2} + \frac{7}{360}}, {b}^{2}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{31}{15120}, {b}^{2}, \frac{7}{360}\right)}, {b}^{2}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, \color{blue}{b \cdot b}, \frac{7}{360}\right), {b}^{2}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, \color{blue}{b \cdot b}, \frac{7}{360}\right), {b}^{2}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, b \cdot b, \frac{7}{360}\right), \color{blue}{b \cdot b}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, b \cdot b, \frac{7}{360}\right), \color{blue}{b \cdot b}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, b \cdot b, \frac{7}{360}\right), b \cdot b, \frac{1}{6}\right), \color{blue}{b \cdot b}, 1\right)}{b}} \]

      15. lower-*.f64 98.6

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), \color{blue}{b \cdot b}, 1\right)}{b}} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}} \]

   if 17 < 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-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 61.7

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 13.1

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

   7. Applied rewrites 13.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.6%

       \[\leadsto r \cdot \left(-1 \cdot \left(-\sin b\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 17:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.85)
   (* (* -1.0 r) (sin b))
   (if (<= b 40.0)
     (/
      (/ r (cos (+ a b)))
      (/
       (fma (fma 0.019444444444444445 (* b b) 0.16666666666666666) (* b b) 1.0)
       b))
     (* (- r) (* -1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.85) {
		tmp = (-1.0 * r) * sin(b);
	} else if (b <= 40.0) {
		tmp = (r / cos((a + b))) / (fma(fma(0.019444444444444445, (b * b), 0.16666666666666666), (b * b), 1.0) / b);
	} else {
		tmp = -r * (-1.0 * sin(b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.85)
		tmp = Float64(Float64(-1.0 * r) * sin(b));
	elseif (b <= 40.0)
		tmp = Float64(Float64(r / cos(Float64(a + b))) / Float64(fma(fma(0.019444444444444445, Float64(b * b), 0.16666666666666666), Float64(b * b), 1.0) / b));
	else
		tmp = Float64(Float64(-r) * Float64(-1.0 * sin(b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.85], N[(N[(-1.0 * r), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 40.0], N[(N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.019444444444444445 * N[(b * b), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8500000000000001

   1. Initial program 57.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-div N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-neg.f64 N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-pow N/A

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

      19. pow2 N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqr-neg N/A

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

   9. Applied rewrites 12.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]
   11. Step-by-step derivation

   12. Applied rewrites 13.8%

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]

   if -1.8500000000000001 < b < 40

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{1 + {b}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {b}^{2}\right)}{b}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{1 + {b}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {b}^{2}\right)}{b}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{{b}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {b}^{2}\right) + 1}}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {b}^{2}\right) \cdot {b}^{2}} + 1}{b}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {b}^{2}, {b}^{2}, 1\right)}}{b}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {b}^{2} + \frac{1}{6}}, {b}^{2}, 1\right)}{b}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {b}^{2}, \frac{1}{6}\right)}, {b}^{2}, 1\right)}{b}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{b \cdot b}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{b \cdot b}, \frac{1}{6}\right), {b}^{2}, 1\right)}{b}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, b \cdot b, \frac{1}{6}\right), \color{blue}{b \cdot b}, 1\right)}{b}} \]

      10. lower-*.f64 98.4

          \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), \color{blue}{b \cdot b}, 1\right)}{b}} \]

   7. Applied rewrites 98.4%

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}} \]

   if 40 < 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-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 61.7

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 13.1

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

   7. Applied rewrites 13.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.6%

       \[\leadsto r \cdot \left(-1 \cdot \left(-\sin b\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 47:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.85)
   (* (* -1.0 r) (sin b))
   (if (<= b 47.0)
     (/ (/ r (cos (+ a b))) (/ (fma (* b b) 0.16666666666666666 1.0) b))
     (* (- r) (* -1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.85) {
		tmp = (-1.0 * r) * sin(b);
	} else if (b <= 47.0) {
		tmp = (r / cos((a + b))) / (fma((b * b), 0.16666666666666666, 1.0) / b);
	} else {
		tmp = -r * (-1.0 * sin(b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.85)
		tmp = Float64(Float64(-1.0 * r) * sin(b));
	elseif (b <= 47.0)
		tmp = Float64(Float64(r / cos(Float64(a + b))) / Float64(fma(Float64(b * b), 0.16666666666666666, 1.0) / b));
	else
		tmp = Float64(Float64(-r) * Float64(-1.0 * sin(b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.85], N[(N[(-1.0 * r), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 47.0], N[(N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8500000000000001

   1. Initial program 57.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-div N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-neg.f64 N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-pow N/A

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

      19. pow2 N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqr-neg N/A

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

   9. Applied rewrites 12.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]
   11. Step-by-step derivation

   12. Applied rewrites 13.8%

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]

   if -1.8500000000000001 < b < 47

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if 47 < 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-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 61.7

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 13.1

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

   7. Applied rewrites 13.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.6%

       \[\leadsto r \cdot \left(-1 \cdot \left(-\sin b\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 47:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -270:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 1.45:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -270.0)
   (* (* -1.0 r) (sin b))
   (if (<= b 1.45) (* r (/ b (cos a))) (* (- r) (* -1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -270.0) {
		tmp = (-1.0 * r) * sin(b);
	} else if (b <= 1.45) {
		tmp = r * (b / cos(a));
	} else {
		tmp = -r * (-1.0 * sin(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 <= (-270.0d0)) then
        tmp = ((-1.0d0) * r) * sin(b)
    else if (b <= 1.45d0) then
        tmp = r * (b / cos(a))
    else
        tmp = -r * ((-1.0d0) * sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -270.0) {
		tmp = (-1.0 * r) * Math.sin(b);
	} else if (b <= 1.45) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = -r * (-1.0 * Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -270.0:
		tmp = (-1.0 * r) * math.sin(b)
	elif b <= 1.45:
		tmp = r * (b / math.cos(a))
	else:
		tmp = -r * (-1.0 * math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -270.0)
		tmp = Float64(Float64(-1.0 * r) * sin(b));
	elseif (b <= 1.45)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(Float64(-r) * Float64(-1.0 * sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -270.0)
		tmp = (-1.0 * r) * sin(b);
	elseif (b <= 1.45)
		tmp = r * (b / cos(a));
	else
		tmp = -r * (-1.0 * sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -270.0], N[(N[(-1.0 * r), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -270

   1. Initial program 57.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-div N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-neg.f64 N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-pow N/A

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

      19. pow2 N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqr-neg N/A

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

   9. Applied rewrites 12.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]
   11. Step-by-step derivation

   12. Applied rewrites 13.8%

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]

   if -270 < b < 1.44999999999999996

   1. Initial program 99.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   if 1.44999999999999996 < 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-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 61.7

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 13.1

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

   7. Applied rewrites 13.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.6%

       \[\leadsto r \cdot \left(-1 \cdot \left(-\sin b\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -270:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 1.45:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -270:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 1.45:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -270.0)
   (* (* -1.0 r) (sin b))
   (if (<= b 1.45) (* (/ r (cos a)) b) (* (- r) (* -1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -270.0) {
		tmp = (-1.0 * r) * sin(b);
	} else if (b <= 1.45) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = -r * (-1.0 * sin(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 <= (-270.0d0)) then
        tmp = ((-1.0d0) * r) * sin(b)
    else if (b <= 1.45d0) then
        tmp = (r / cos(a)) * b
    else
        tmp = -r * ((-1.0d0) * sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -270.0) {
		tmp = (-1.0 * r) * Math.sin(b);
	} else if (b <= 1.45) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = -r * (-1.0 * Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -270.0:
		tmp = (-1.0 * r) * math.sin(b)
	elif b <= 1.45:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = -r * (-1.0 * math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -270.0)
		tmp = Float64(Float64(-1.0 * r) * sin(b));
	elseif (b <= 1.45)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = Float64(Float64(-r) * Float64(-1.0 * sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -270.0)
		tmp = (-1.0 * r) * sin(b);
	elseif (b <= 1.45)
		tmp = (r / cos(a)) * b;
	else
		tmp = -r * (-1.0 * sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -270.0], N[(N[(-1.0 * r), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -270

   1. Initial program 57.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-div N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-neg.f64 N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-pow N/A

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

      19. pow2 N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqr-neg N/A

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

   9. Applied rewrites 12.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]
   11. Step-by-step derivation

   12. Applied rewrites 13.8%

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]

   if -270 < b < 1.44999999999999996

   1. Initial program 99.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if 1.44999999999999996 < 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-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 61.7

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 13.1

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

   7. Applied rewrites 13.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.6%

       \[\leadsto r \cdot \left(-1 \cdot \left(-\sin b\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -270:\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{elif}\;b \leq 1.45:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-6} \lor \neg \left(b \leq 96000\right):\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.3e-6) (not (<= b 96000.0)))
   (* (* -1.0 r) (sin b))
   (* r (/ b 1.0))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.3e-6) || !(b <= 96000.0)) {
		tmp = (-1.0 * r) * sin(b);
	} else {
		tmp = r * (b / 1.0);
	}
	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) :: tmp
    if ((b <= (-1.3d-6)) .or. (.not. (b <= 96000.0d0))) then
        tmp = ((-1.0d0) * r) * sin(b)
    else
        tmp = r * (b / 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.3e-6) || !(b <= 96000.0)) {
		tmp = (-1.0 * r) * Math.sin(b);
	} else {
		tmp = r * (b / 1.0);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.3e-6) or not (b <= 96000.0):
		tmp = (-1.0 * r) * math.sin(b)
	else:
		tmp = r * (b / 1.0)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.3e-6) || !(b <= 96000.0))
		tmp = Float64(Float64(-1.0 * r) * sin(b));
	else
		tmp = Float64(r * Float64(b / 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.3e-6) || ~((b <= 96000.0)))
		tmp = (-1.0 * r) * sin(b);
	else
		tmp = r * (b / 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.3e-6], N[Not[LessEqual[b, 96000.0]], $MachinePrecision]], N[(N[(-1.0 * r), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.30000000000000005e-6 or 96000 < b

   1. Initial program 59.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 12.7

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

   7. Applied rewrites 12.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-div N/A

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

      7. unpow-1 N/A

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

      8. lift-pow.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-neg.f64 N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-pow N/A

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

      19. pow2 N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqr-neg N/A

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

   9. Applied rewrites 11.6%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]
   11. Step-by-step derivation

   12. Applied rewrites 12.1%

       \[\leadsto \left(-1 \cdot r\right) \cdot \sin b \]

   if -1.30000000000000005e-6 < b < 96000

   1. Initial program 99.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.8%

      \[\leadsto r \cdot \frac{b}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-6} \lor \neg \left(b \leq 96000\right):\\ \;\;\;\;\left(-1 \cdot r\right) \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(-r\right) \cdot \left(-1 \cdot \sin b\right) \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (- r) (* -1.0 (sin b))))

C

double code(double r, double a, double b) {
	return -r * (-1.0 * 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 * ((-1.0d0) * sin(b))
end function

Java

public static double code(double r, double a, double b) {
	return -r * (-1.0 * Math.sin(b));
}

Python

def code(r, a, b):
	return -r * (-1.0 * math.sin(b))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 79.7

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

4. Applied rewrites 79.7%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 56.1

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

7. Applied rewrites 56.1%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 36.9%

    \[\leadsto r \cdot \left(-1 \cdot \left(-\sin b\right)\right) \]

11. Final simplification 36.9%

    \[\leadsto \left(-r\right) \cdot \left(-1 \cdot \sin b\right) \]
12. Add Preprocessing

Alternative 15: 34.9% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

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 / 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 51.7

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.5%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))