rsin B (should all be same)

Percentage Accurate: 76.9% → 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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ t_1 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))) (t_1 (* (/ r (cos b)) (sin b))))
   (if (<= t_0 -1e-6) t_1 (if (<= t_0 1e-7) (* (/ b (cos a)) r) t_1))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double t_1 = (r / cos(b)) * sin(b);
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_1;
	} else if (t_0 <= 1e-7) {
		tmp = (b / cos(a)) * r;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    t_1 = (r / cos(b)) * sin(b)
    if (t_0 <= (-1d-6)) then
        tmp = t_1
    else if (t_0 <= 1d-7) then
        tmp = (b / cos(a)) * r
    else
        tmp = t_1
    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 t_1 = (r / Math.cos(b)) * Math.sin(b);
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_1;
	} else if (t_0 <= 1e-7) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	t_1 = (r / math.cos(b)) * math.sin(b)
	tmp = 0
	if t_0 <= -1e-6:
		tmp = t_1
	elif t_0 <= 1e-7:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_1
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	t_1 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (t_0 <= -1e-6)
		tmp = t_1;
	elseif (t_0 <= 1e-7)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((a + b));
	t_1 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (t_0 <= -1e-6)
		tmp = t_1;
	elseif (t_0 <= 1e-7)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-6], t$95$1, If[LessEqual[t$95$0, 1e-7], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -9.99999999999999955e-7 or 9.9999999999999995e-8 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 50.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 50.2

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

   5. Applied rewrites 50.2%

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

   if -9.99999999999999955e-7 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 9.9999999999999995e-8

   1. Initial program 99.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 10^{-7}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. sin-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sin-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. 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. Final simplification 99.5%

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

Alternative 6: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos a} \cdot r\\ \mathbf{if}\;a \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 50:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) (cos a)) r)))
   (if (<= a -5e-6) t_0 (if (<= a 50.0) (* (/ (sin b) (cos b)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / cos(a)) * r;
	double tmp;
	if (a <= -5e-6) {
		tmp = t_0;
	} else if (a <= 50.0) {
		tmp = (sin(b) / cos(b)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / cos(a)) * r
    if (a <= (-5d-6)) then
        tmp = t_0
    else if (a <= 50.0d0) then
        tmp = (sin(b) / cos(b)) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / Math.cos(a)) * r;
	double tmp;
	if (a <= -5e-6) {
		tmp = t_0;
	} else if (a <= 50.0) {
		tmp = (Math.sin(b) / Math.cos(b)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / math.cos(a)) * r
	tmp = 0
	if a <= -5e-6:
		tmp = t_0
	elif a <= 50.0:
		tmp = (math.sin(b) / math.cos(b)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / cos(a)) * r)
	tmp = 0.0
	if (a <= -5e-6)
		tmp = t_0;
	elseif (a <= 50.0)
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / cos(a)) * r;
	tmp = 0.0;
	if (a <= -5e-6)
		tmp = t_0;
	elseif (a <= 50.0)
		tmp = (sin(b) / cos(b)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -5e-6], t$95$0, If[LessEqual[a, 50.0], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000041e-6 or 50 < a

   1. Initial program 54.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

   if -5.00000000000000041e-6 < a < 50

   1. Initial program 97.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 50:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos a} \cdot r\\ \mathbf{if}\;a \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 50:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) (cos a)) r)))
   (if (<= a -5e-6) t_0 (if (<= a 50.0) (* (/ r (cos b)) (sin b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / cos(a)) * r;
	double tmp;
	if (a <= -5e-6) {
		tmp = t_0;
	} else if (a <= 50.0) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / cos(a)) * r
    if (a <= (-5d-6)) then
        tmp = t_0
    else if (a <= 50.0d0) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / Math.cos(a)) * r;
	double tmp;
	if (a <= -5e-6) {
		tmp = t_0;
	} else if (a <= 50.0) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / math.cos(a)) * r
	tmp = 0
	if a <= -5e-6:
		tmp = t_0
	elif a <= 50.0:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / cos(a)) * r)
	tmp = 0.0
	if (a <= -5e-6)
		tmp = t_0;
	elseif (a <= 50.0)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / cos(a)) * r;
	tmp = 0.0;
	if (a <= -5e-6)
		tmp = t_0;
	elseif (a <= 50.0)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -5e-6], t$95$0, If[LessEqual[a, 50.0], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000041e-6 or 50 < a

   1. Initial program 54.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

   if -5.00000000000000041e-6 < a < 50

   1. Initial program 97.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 50:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Final simplification 75.0%

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

Alternative 9: 55.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{if}\;b \leq -4.25 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 10:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (- r) (* -1.0 (sin b)))))
   (if (<= b -4.25e+18)
     t_0
     (if (<= b 10.0)
       (/
        (/ r (cos (+ a b)))
        (/
         (fma
          (fma
           (fma 0.00205026455026455 (* b b) 0.019444444444444445)
           (* b b)
           0.16666666666666666)
          (* b b)
          1.0)
         b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = -r * (-1.0 * sin(b));
	double tmp;
	if (b <= -4.25e+18) {
		tmp = t_0;
	} else if (b <= 10.0) {
		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 = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(-r) * Float64(-1.0 * sin(b)))
	tmp = 0.0
	if (b <= -4.25e+18)
		tmp = t_0;
	elseif (b <= 10.0)
		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 = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.25e+18], t$95$0, If[LessEqual[b, 10.0], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.25e18 or 10 < b

   1. Initial program 49.0%

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

      1. lift-/.f64 N/A

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

      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 48.9

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 48.9

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

   7. Applied rewrites 48.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 13.5%

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

   if -4.25e18 < b < 10

   1. Initial program 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. 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 97.0

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

   4. Applied rewrites 97.0%

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

   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 94.6

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

   7. Applied rewrites 94.6%

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.25 \cdot 10^{+18}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{elif}\;b \leq 10:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, b \cdot b, 0.019444444444444445\right), b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{if}\;b \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 10:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (- r) (* -1.0 (sin b)))))
   (if (<= b -5.4e+18)
     t_0
     (if (<= b 10.0)
       (/
        (/ r (cos (+ a b)))
        (/
         (fma
          (fma 0.019444444444444445 (* b b) 0.16666666666666666)
          (* b b)
          1.0)
         b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = -r * (-1.0 * sin(b));
	double tmp;
	if (b <= -5.4e+18) {
		tmp = t_0;
	} else if (b <= 10.0) {
		tmp = (r / cos((a + b))) / (fma(fma(0.019444444444444445, (b * b), 0.16666666666666666), (b * b), 1.0) / b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(-r) * Float64(-1.0 * sin(b)))
	tmp = 0.0
	if (b <= -5.4e+18)
		tmp = t_0;
	elseif (b <= 10.0)
		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 = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.4e+18], t$95$0, If[LessEqual[b, 10.0], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.4e18 or 10 < b

   1. Initial program 49.0%

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

      1. lift-/.f64 N/A

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

      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 48.9

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 48.9

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

   7. Applied rewrites 48.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 13.5%

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

   if -5.4e18 < b < 10

   1. Initial program 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. 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 97.0

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

   4. Applied rewrites 97.0%

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

   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 94.5

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

   7. Applied rewrites 94.5%

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{elif}\;b \leq 10:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, b \cdot b, 0.16666666666666666\right), b \cdot b, 1\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -5.4e+18)
   (* (- r) (* -1.0 (sin b)))
   (/ (/ r (cos (+ a b))) (/ (fma (* b b) 0.16666666666666666 1.0) b))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -5.4e+18) {
		tmp = -r * (-1.0 * sin(b));
	} else {
		tmp = (r / cos((a + b))) / (fma((b * b), 0.16666666666666666, 1.0) / b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -5.4e+18)
		tmp = Float64(Float64(-r) * Float64(-1.0 * sin(b)));
	else
		tmp = Float64(Float64(r / cos(Float64(a + b))) / Float64(fma(Float64(b * b), 0.16666666666666666, 1.0) / b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -5.4e+18], N[((-r) * N[(-1.0 * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b < -5.4e18

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      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 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 50.9

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

   7. Applied rewrites 50.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 13.9%

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

   if -5.4e18 < b

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. 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 83.9

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

   4. Applied rewrites 83.9%

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

   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 71.0

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

   7. Applied rewrites 71.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -r * ((-1.0d0) * sin(b))
    if (b <= (-1.0d0)) then
        tmp = t_0
    else if (b <= 4.6d0) then
        tmp = (b / cos(a)) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1 or 4.5999999999999996 < b

   1. Initial program 49.3%

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

      1. lift-/.f64 N/A

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

      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 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 49.0

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 13.1%

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

   if -1 < b < 4.5999999999999996

   1. Initial program 99.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{elif}\;b \leq 4.6:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -r * ((-1.0d0) * sin(b))
    if (b <= (-1.0d0)) then
        tmp = t_0
    else if (b <= 4.6d0) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1 or 4.5999999999999996 < b

   1. Initial program 49.3%

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

      1. lift-/.f64 N/A

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

      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 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 49.0

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 13.1%

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

   if -1 < b < 4.5999999999999996

   1. Initial program 99.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \mathbf{elif}\;b \leq 4.6:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-r\right) \cdot \left(-1 \cdot \sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.3% 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 75.0%

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

   1. lift-/.f64 N/A

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

   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 74.9

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

4. Applied rewrites 74.9%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 58.4

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

7. Applied rewrites 58.4%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 40.4%

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

11. Final simplification 40.4%

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

Alternative 15: 35.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.9%

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

9. Final simplification 35.9%

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

Reproduce

herbie shell --seed 2024325 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))