rsin A (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 12.7s

Alternatives: 16

Speedup: 0.4×

Specification

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 78.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 78.9

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

4. Applied rewrites 78.9%

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

   1. cos-sum N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-neg N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 78.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 78.9

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

4. Applied rewrites 78.9%

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

   1. +-commutative N/A

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

   2. cos-sum N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -3.3e-6)
   (/ (* r (sin b)) (cos a))
   (if (<= a 1.46e-5) (* r (tan b)) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.3e-6) {
		tmp = (r * sin(b)) / cos(a);
	} else if (a <= 1.46e-5) {
		tmp = r * tan(b);
	} else {
		tmp = sin(b) * (r / 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) :: tmp
    if (a <= (-3.3d-6)) then
        tmp = (r * sin(b)) / cos(a)
    else if (a <= 1.46d-5) then
        tmp = r * tan(b)
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.3e-6) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else if (a <= 1.46e-5) {
		tmp = r * Math.tan(b);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -3.3e-6:
		tmp = (r * math.sin(b)) / math.cos(a)
	elif a <= 1.46e-5:
		tmp = r * math.tan(b)
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -3.3e-6)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	elseif (a <= 1.46e-5)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -3.3e-6)
		tmp = (r * sin(b)) / cos(a);
	elseif (a <= 1.46e-5)
		tmp = r * tan(b);
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -3.3e-6], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.46e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.30000000000000017e-6

   1. Initial program 51.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 52.3

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

   5. Applied rewrites 52.3%

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

   if -3.30000000000000017e-6 < a < 1.46000000000000008e-5

   1. Initial program 99.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 99.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if 1.46000000000000008e-5 < a

   1. Initial program 65.0%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 65.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 65.1

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

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 64.9

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

   7. Applied rewrites 64.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -1e-5) t_0 (if (<= a 1.46e-5) (* r (tan b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(a));
	double tmp;
	if (a <= -1e-5) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * tan(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) * (r / cos(a))
    if (a <= (-1d-5)) then
        tmp = t_0
    else if (a <= 1.46d-5) then
        tmp = r * tan(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) * (r / Math.cos(a));
	double tmp;
	if (a <= -1e-5) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * Math.tan(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(a))
	tmp = 0
	if a <= -1e-5:
		tmp = t_0
	elif a <= 1.46e-5:
		tmp = r * math.tan(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(a)))
	tmp = 0.0
	if (a <= -1e-5)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = Float64(r * tan(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(a));
	tmp = 0.0;
	if (a <= -1e-5)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = r * tan(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1e-5], t$95$0, If[LessEqual[a, 1.46e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000008e-5 or 1.46000000000000008e-5 < a

   1. Initial program 58.2%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 58.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -1.00000000000000008e-5 < a < 1.46000000000000008e-5

   1. Initial program 99.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 99.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 99.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -1e-5) t_0 (if (<= a 1.46e-5) (* r (tan b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -1e-5) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * tan(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 * (sin(b) / cos(a))
    if (a <= (-1d-5)) then
        tmp = t_0
    else if (a <= 1.46d-5) then
        tmp = r * tan(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 * (Math.sin(b) / Math.cos(a));
	double tmp;
	if (a <= -1e-5) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * Math.tan(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -1e-5:
		tmp = t_0
	elif a <= 1.46e-5:
		tmp = r * math.tan(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -1e-5)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = Float64(r * tan(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -1e-5)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = r * tan(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1e-5], t$95$0, If[LessEqual[a, 1.46e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000008e-5 or 1.46000000000000008e-5 < a

   1. Initial program 58.2%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 58.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 58.7

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

   7. Applied rewrites 58.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 58.6

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

   9. Applied rewrites 58.6%

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

   if -1.00000000000000008e-5 < a < 1.46000000000000008e-5

   1. Initial program 99.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 99.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 99.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 78.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 78.9

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

4. Applied rewrites 78.9%

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

5. Final simplification 78.9%

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

Alternative 7: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.9:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.202:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.9)
   (* (tan b) (/ 1.0 (/ 1.0 r)))
   (if (<= b 0.202)
     (/
      (*
       r
       (fma
        (fma
         b
         (* b (fma b (* b -0.0001984126984126984) 0.008333333333333333))
         -0.16666666666666666)
        (* b (* b b))
        b))
      (cos (+ b a)))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.9) {
		tmp = tan(b) * (1.0 / (1.0 / r));
	} else if (b <= 0.202) {
		tmp = (r * fma(fma(b, (b * fma(b, (b * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b)) / cos((b + a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.9)
		tmp = Float64(tan(b) * Float64(1.0 / Float64(1.0 / r)));
	elseif (b <= 0.202)
		tmp = Float64(Float64(r * fma(fma(b, Float64(b * fma(b, Float64(b * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b)) / cos(Float64(b + a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.9], N[(N[Tan[b], $MachinePrecision] * N[(1.0 / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.202], N[(N[(r * N[(N[(b * N[(b * N[(b * N[(b * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.900000000000000022

   1. Initial program 65.1%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 65.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      14. lift-tan.f64 N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan b \cdot \frac{-1 \cdot -1}{\frac{1}{r}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \color{blue}{\frac{-1 \cdot -1}{\frac{1}{r}}} \]

      17. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{\color{blue}{1}}{\frac{1}{r}} \]

      18. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{-1 \cdot -1}}{r}} \]

      19. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \frac{1}{\color{blue}{\frac{-1 \cdot -1}{r}}} \]

      20. metadata-eval 63.0

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{1}}{r}} \]

   9. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\tan b \cdot \frac{1}{\frac{1}{r}}} \]

   if -0.900000000000000022 < b < 0.20200000000000001

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 97.6%

      \[\leadsto \frac{r \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(a + b\right)} \]

   if 0.20200000000000001 < b

   1. Initial program 51.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 53.1

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

   7. Applied rewrites 53.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 53.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 53.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.9:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.202:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.9:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.052:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.9)
   (* (tan b) (/ 1.0 (/ 1.0 r)))
   (if (<= b 0.052)
     (/
      (*
       b
       (fma
        (* b b)
        (* r (fma (* b b) 0.008333333333333333 -0.16666666666666666))
        r))
      (cos (+ b a)))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.9) {
		tmp = tan(b) * (1.0 / (1.0 / r));
	} else if (b <= 0.052) {
		tmp = (b * fma((b * b), (r * fma((b * b), 0.008333333333333333, -0.16666666666666666)), r)) / cos((b + a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.9)
		tmp = Float64(tan(b) * Float64(1.0 / Float64(1.0 / r)));
	elseif (b <= 0.052)
		tmp = Float64(Float64(b * fma(Float64(b * b), Float64(r * fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666)), r)) / cos(Float64(b + a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.9], N[(N[Tan[b], $MachinePrecision] * N[(1.0 / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.052], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(r * N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.900000000000000022

   1. Initial program 65.1%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 65.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      14. lift-tan.f64 N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan b \cdot \frac{-1 \cdot -1}{\frac{1}{r}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \color{blue}{\frac{-1 \cdot -1}{\frac{1}{r}}} \]

      17. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{\color{blue}{1}}{\frac{1}{r}} \]

      18. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{-1 \cdot -1}}{r}} \]

      19. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \frac{1}{\color{blue}{\frac{-1 \cdot -1}{r}}} \]

      20. metadata-eval 63.0

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{1}}{r}} \]

   9. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\tan b \cdot \frac{1}{\frac{1}{r}}} \]

   if -0.900000000000000022 < b < 0.0519999999999999976

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 0.0519999999999999976 < b

   1. Initial program 51.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 53.1

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

   7. Applied rewrites 53.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 53.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 53.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.9:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.052:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.005:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.0078:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.005)
   (* (tan b) (/ 1.0 (/ 1.0 r)))
   (if (<= b 0.0078)
     (/ (* r (fma (* b b) (* b -0.16666666666666666) b)) (cos (+ b a)))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.005) {
		tmp = tan(b) * (1.0 / (1.0 / r));
	} else if (b <= 0.0078) {
		tmp = (r * fma((b * b), (b * -0.16666666666666666), b)) / cos((b + a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.005)
		tmp = Float64(tan(b) * Float64(1.0 / Float64(1.0 / r)));
	elseif (b <= 0.0078)
		tmp = Float64(Float64(r * fma(Float64(b * b), Float64(b * -0.16666666666666666), b)) / cos(Float64(b + a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.005], N[(N[Tan[b], $MachinePrecision] * N[(1.0 / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0078], N[(N[(r * N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0050000000000000001

   1. Initial program 64.9%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 64.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 62.5

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

   7. Applied rewrites 62.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      14. lift-tan.f64 N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan b \cdot \frac{-1 \cdot -1}{\frac{1}{r}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \color{blue}{\frac{-1 \cdot -1}{\frac{1}{r}}} \]

      17. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{\color{blue}{1}}{\frac{1}{r}} \]

      18. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{-1 \cdot -1}}{r}} \]

      19. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \frac{1}{\color{blue}{\frac{-1 \cdot -1}{r}}} \]

      20. metadata-eval 62.6

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{1}}{r}} \]

   9. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\tan b \cdot \frac{1}{\frac{1}{r}}} \]

   if -0.0050000000000000001 < b < 0.0077999999999999996

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.0077999999999999996 < b

   1. Initial program 51.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 53.1

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

   7. Applied rewrites 53.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 53.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 53.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.005:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.0078:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.005:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.0078:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.005)
   (* (tan b) (/ 1.0 (/ 1.0 r)))
   (if (<= b 0.0078)
     (/ (* b (fma -0.16666666666666666 (* r (* b b)) r)) (cos (+ b a)))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.005) {
		tmp = tan(b) * (1.0 / (1.0 / r));
	} else if (b <= 0.0078) {
		tmp = (b * fma(-0.16666666666666666, (r * (b * b)), r)) / cos((b + a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.005)
		tmp = Float64(tan(b) * Float64(1.0 / Float64(1.0 / r)));
	elseif (b <= 0.0078)
		tmp = Float64(Float64(b * fma(-0.16666666666666666, Float64(r * Float64(b * b)), r)) / cos(Float64(b + a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.005], N[(N[Tan[b], $MachinePrecision] * N[(1.0 / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0078], N[(N[(b * N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0050000000000000001

   1. Initial program 64.9%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 64.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 62.5

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

   7. Applied rewrites 62.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      14. lift-tan.f64 N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan b \cdot \frac{-1 \cdot -1}{\frac{1}{r}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \color{blue}{\frac{-1 \cdot -1}{\frac{1}{r}}} \]

      17. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{\color{blue}{1}}{\frac{1}{r}} \]

      18. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{-1 \cdot -1}}{r}} \]

      19. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \frac{1}{\color{blue}{\frac{-1 \cdot -1}{r}}} \]

      20. metadata-eval 62.6

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{1}}{r}} \]

   9. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\tan b \cdot \frac{1}{\frac{1}{r}}} \]

   if -0.0050000000000000001 < b < 0.0077999999999999996

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.0077999999999999996 < b

   1. Initial program 51.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 53.1

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

   7. Applied rewrites 53.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 53.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 53.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.005:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 0.0078:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00023:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.00023)
   (* (tan b) (/ 1.0 (/ 1.0 r)))
   (if (<= b 1.35e-5) (/ (* r b) (cos (+ b a))) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.00023) {
		tmp = tan(b) * (1.0 / (1.0 / r));
	} else if (b <= 1.35e-5) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = r * tan(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 <= (-0.00023d0)) then
        tmp = tan(b) * (1.0d0 / (1.0d0 / r))
    else if (b <= 1.35d-5) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.00023) {
		tmp = Math.tan(b) * (1.0 / (1.0 / r));
	} else if (b <= 1.35e-5) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.00023:
		tmp = math.tan(b) * (1.0 / (1.0 / r))
	elif b <= 1.35e-5:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.00023)
		tmp = Float64(tan(b) * Float64(1.0 / Float64(1.0 / r)));
	elseif (b <= 1.35e-5)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.00023)
		tmp = tan(b) * (1.0 / (1.0 / r));
	elseif (b <= 1.35e-5)
		tmp = (r * b) / cos((b + a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.00023], N[(N[Tan[b], $MachinePrecision] * N[(1.0 / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-5], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3000000000000001e-4

   1. Initial program 64.9%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 64.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 62.5

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

   7. Applied rewrites 62.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      14. lift-tan.f64 N/A

          \[\leadsto \color{blue}{\tan b} \cdot \frac{-1 \cdot -1}{\frac{1}{r}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan b \cdot \frac{-1 \cdot -1}{\frac{1}{r}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \color{blue}{\frac{-1 \cdot -1}{\frac{1}{r}}} \]

      17. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{\color{blue}{1}}{\frac{1}{r}} \]

      18. metadata-eval N/A

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{-1 \cdot -1}}{r}} \]

      19. lower-/.f64 N/A

          \[\leadsto \tan b \cdot \frac{1}{\color{blue}{\frac{-1 \cdot -1}{r}}} \]

      20. metadata-eval 62.6

          \[\leadsto \tan b \cdot \frac{1}{\frac{\color{blue}{1}}{r}} \]

   9. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\tan b \cdot \frac{1}{\frac{1}{r}}} \]

   if -2.3000000000000001e-4 < b < 1.3499999999999999e-5

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 1.3499999999999999e-5 < b

   1. Initial program 51.3%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 53.1

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

   7. Applied rewrites 53.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 53.2

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 53.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00023:\\ \;\;\;\;\tan b \cdot \frac{1}{\frac{1}{r}}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.00023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.00023) t_0 (if (<= b 1.35e-5) (/ (* r b) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.00023) {
		tmp = t_0;
	} else if (b <= 1.35e-5) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-0.00023d0)) then
        tmp = t_0
    else if (b <= 1.35d-5) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -0.00023) {
		tmp = t_0;
	} else if (b <= 1.35e-5) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -0.00023:
		tmp = t_0
	elif b <= 1.35e-5:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -0.00023)
		tmp = t_0;
	elseif (b <= 1.35e-5)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -0.00023)
		tmp = t_0;
	elseif (b <= 1.35e-5)
		tmp = (r * b) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.00023], t$95$0, If[LessEqual[b, 1.35e-5], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3000000000000001e-4 or 1.3499999999999999e-5 < b

   1. Initial program 57.8%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 57.8

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 57.6

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

   7. Applied rewrites 57.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 57.7

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -2.3000000000000001e-4 < b < 1.3499999999999999e-5

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00023:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.00023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.00023) t_0 (if (<= b 1.35e-5) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.00023) {
		tmp = t_0;
	} else if (b <= 1.35e-5) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-0.00023d0)) then
        tmp = t_0
    else if (b <= 1.35d-5) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -0.00023) {
		tmp = t_0;
	} else if (b <= 1.35e-5) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -0.00023:
		tmp = t_0
	elif b <= 1.35e-5:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -0.00023)
		tmp = t_0;
	elseif (b <= 1.35e-5)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -0.00023)
		tmp = t_0;
	elseif (b <= 1.35e-5)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.00023], t$95$0, If[LessEqual[b, 1.35e-5], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3000000000000001e-4 or 1.3499999999999999e-5 < b

   1. Initial program 57.8%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 57.8

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 57.6

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

   7. Applied rewrites 57.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      8. lift-tan.f64 N/A

         \[\leadsto r \cdot \color{blue}{\tan b} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      10. lower-*.f64 57.7

          \[\leadsto \color{blue}{\tan b \cdot r} \]

   9. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -2.3000000000000001e-4 < b < 1.3499999999999999e-5

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 97.9

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

   5. Applied rewrites 97.9%

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

      1. *-commutative N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 97.9

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

   7. Applied rewrites 97.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00023:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 78.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 78.9

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

4. Applied rewrites 78.9%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 61.3

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

7. Applied rewrites 61.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. tan-quot N/A

      \[\leadsto r \cdot \color{blue}{\tan b} \]

   8. lift-tan.f64 N/A

      \[\leadsto r \cdot \color{blue}{\tan b} \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   10. lower-*.f64 61.3

       \[\leadsto \color{blue}{\tan b \cdot r} \]

9. Applied rewrites 61.3%

   \[\leadsto \color{blue}{\tan b \cdot r} \]

10. Final simplification 61.3%

    \[\leadsto r \cdot \tan b \]
11. Add Preprocessing

Alternative 15: 39.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 54.9

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

5. Applied rewrites 54.9%

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

6. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 38.9

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

8. Applied rewrites 38.9%

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

Alternative 16: 35.4% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{r \cdot b} \]

   2. lower-*.f64 35.6

      \[\leadsto \color{blue}{r \cdot b} \]

8. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))