rsin B (should all be same)

Percentage Accurate: 77.2% → 99.5%

Time: 12.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 80.9

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

4. Applied rewrites 80.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\cos \left(a - b\right) \cdot \left(r \cdot \sin b\right)}{\cos \left(b - \left(a - \left(a + b\right)\right)\right) + \cos \left(\left(\left(b - a\right) - b\right) - a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (*
  2.0
  (/
   (* (cos (- a b)) (* r (sin b)))
   (+ (cos (- b (- a (+ a b)))) (cos (- (- (- b a) b) a))))))

C

double code(double r, double a, double b) {
	return 2.0 * ((cos((a - b)) * (r * sin(b))) / (cos((b - (a - (a + b)))) + cos((((b - a) - 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 = 2.0d0 * ((cos((a - b)) * (r * sin(b))) / (cos((b - (a - (a + b)))) + cos((((b - a) - b) - a))))
end function

Java

public static double code(double r, double a, double b) {
	return 2.0 * ((Math.cos((a - b)) * (r * Math.sin(b))) / (Math.cos((b - (a - (a + b)))) + Math.cos((((b - a) - b) - a))));
}

Python

def code(r, a, b):
	return 2.0 * ((math.cos((a - b)) * (r * math.sin(b))) / (math.cos((b - (a - (a + b)))) + math.cos((((b - a) - b) - a))))

Julia

function code(r, a, b)
	return Float64(2.0 * Float64(Float64(cos(Float64(a - b)) * Float64(r * sin(b))) / Float64(cos(Float64(b - Float64(a - Float64(a + b)))) + cos(Float64(Float64(Float64(b - a) - b) - a)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = 2.0 * ((cos((a - b)) * (r * sin(b))) / (cos((b - (a - (a + b)))) + cos((((b - a) - b) - a))));
end

Wolfram

code[r_, a_, b_] := N[(2.0 * N[(N[(N[Cos[N[(a - b), $MachinePrecision]], $MachinePrecision] * N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(b - N[(a - N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[(N[(b - a), $MachinePrecision] - b), $MachinePrecision] - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 80.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 81.3

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

8. Applied rewrites 81.3%

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

9. Final simplification 81.3%

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

Alternative 4: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -0.0011) t_0 (if (<= b 7.2e-8) (* (/ b (cos a)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -0.0011) {
		tmp = t_0;
	} else if (b <= 7.2e-8) {
		tmp = (b / cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r / cos(b)) * sin(b)
    if (b <= (-0.0011d0)) then
        tmp = t_0
    else if (b <= 7.2d-8) then
        tmp = (b / cos(a)) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r / Math.cos(b)) * Math.sin(b);
	double tmp;
	if (b <= -0.0011) {
		tmp = t_0;
	} else if (b <= 7.2e-8) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(b)) * math.sin(b)
	tmp = 0
	if b <= -0.0011:
		tmp = t_0
	elif b <= 7.2e-8:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -0.0011)
		tmp = t_0;
	elseif (b <= 7.2e-8)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (b <= -0.0011)
		tmp = t_0;
	elseif (b <= 7.2e-8)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0011], t$95$0, If[LessEqual[b, 7.2e-8], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00110000000000000007 or 7.19999999999999962e-8 < b

   1. Initial program 55.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

   if -0.00110000000000000007 < b < 7.19999999999999962e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.2% 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]

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 80.9

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

4. Applied rewrites 80.9%

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

5. Final simplification 80.9%

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

Alternative 6: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 80.9

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

4. Applied rewrites 80.9%

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

Alternative 7: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Final simplification 80.9%

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

Alternative 8: 55.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.6:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, b \cdot b, 1\right) \cdot r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -1.05e+14)
     t_0
     (if (<= b 4.6)
       (/ (* (* (fma -0.16666666666666666 (* b b) 1.0) r) b) (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -1.05e+14) {
		tmp = t_0;
	} else if (b <= 4.6) {
		tmp = ((fma(-0.16666666666666666, (b * b), 1.0) * r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -1.05e+14)
		tmp = t_0;
	elseif (b <= 4.6)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(b * b), 1.0) * r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -1.05e+14], t$95$0, If[LessEqual[b, 4.6], N[(N[(N[(N[(-0.16666666666666666 * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e14 or 4.5999999999999996 < b

   1. Initial program 55.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 13.0%

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

   if -1.05e14 < b < 4.5999999999999996

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \mathbf{elif}\;b \leq 4.6:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, b \cdot b, 1\right) \cdot r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 260:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -4.0) t_0 (if (<= b 260.0) (/ (* r b) (cos (+ a b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -4.0) {
		tmp = t_0;
	} else if (b <= 260.0) {
		tmp = (r * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / 1.0d0) * r
    if (b <= (-4.0d0)) then
        tmp = t_0
    else if (b <= 260.0d0) then
        tmp = (r * b) / cos((a + b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / 1.0) * r;
	double tmp;
	if (b <= -4.0) {
		tmp = t_0;
	} else if (b <= 260.0) {
		tmp = (r * b) / Math.cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / 1.0) * r
	tmp = 0
	if b <= -4.0:
		tmp = t_0
	elif b <= 260.0:
		tmp = (r * b) / math.cos((a + b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -4.0)
		tmp = t_0;
	elseif (b <= 260.0)
		tmp = Float64(Float64(r * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / 1.0) * r;
	tmp = 0.0;
	if (b <= -4.0)
		tmp = t_0;
	elseif (b <= 260.0)
		tmp = (r * b) / cos((a + b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -4.0], t$95$0, If[LessEqual[b, 260.0], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4 or 260 < b

   1. Initial program 54.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 6.0

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

   5. Applied rewrites 6.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 12.9%

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

   if -4 < b < 260

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \mathbf{elif}\;b \leq 260:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -4.8) t_0 (if (<= b 2.0) (* (/ b (cos a)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -4.8) {
		tmp = t_0;
	} else if (b <= 2.0) {
		tmp = (b / cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / 1.0d0) * r
    if (b <= (-4.8d0)) then
        tmp = t_0
    else if (b <= 2.0d0) then
        tmp = (b / cos(a)) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / 1.0) * r;
	double tmp;
	if (b <= -4.8) {
		tmp = t_0;
	} else if (b <= 2.0) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / 1.0) * r
	tmp = 0
	if b <= -4.8:
		tmp = t_0
	elif b <= 2.0:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -4.8)
		tmp = t_0;
	elseif (b <= 2.0)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / 1.0) * r;
	tmp = 0.0;
	if (b <= -4.8)
		tmp = t_0;
	elseif (b <= 2.0)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -4.8], t$95$0, If[LessEqual[b, 2.0], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 2 < b

   1. Initial program 55.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 6.0

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

   5. Applied rewrites 6.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 12.8%

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

   if -4.79999999999999982 < b < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \mathbf{elif}\;b \leq 2:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 59.5

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

5. Applied rewrites 59.5%

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

6. Final simplification 59.5%

   \[\leadsto \frac{b}{\cos a} \cdot r \]
7. Add Preprocessing

Alternative 12: 51.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 80.9

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

4. Applied rewrites 80.9%

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

5. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 59.5

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

7. Applied rewrites 59.5%

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

Alternative 13: 35.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 59.5

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

5. Applied rewrites 59.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 37.9%

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

9. Final simplification 37.9%

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

Reproduce

herbie shell --seed 2024283 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))