rsin A (should all be same)

Percentage Accurate: 76.7% → 99.5%

Time: 10.2s

Alternatives: 14

Speedup: 1.0×

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: 76.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-sin.f64 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-sin.f64 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 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 70.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 4: 76.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return ((sin(b) * r) / ((cos((((b - a) - b) - a)) + cos((b - (a - (b + a))))) / 2.0)) * 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) - b) - a)) + cos((b - (a - (b + a))))) / 2.0d0)) * 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) - b) - a)) + Math.cos((b - (a - (b + a))))) / 2.0)) * Math.cos((b - a));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. cos-sum N/A

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

   5. flip-- N/A

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

   6. *-commutative N/A

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

   7. lift-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-sin.f64 N/A

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

   10. cos-diff-rev N/A

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

   11. associate-/r/ N/A

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

   12. lower-*.f64 N/A

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

4. Applied rewrites 70.5%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-cos.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. sin-+PI/2-rev N/A

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

   16. lift-sin.f64 N/A

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

   17. cos-+PI/2-rev N/A

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

   18. lift-sin.f64 N/A

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

   19. sin-diff-rev N/A

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

   20. lower-sin.f64 N/A

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

   21. lower--.f64 N/A

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

6. Applied rewrites 38.3%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

8. Applied rewrites 71.2%

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

Alternative 5: 76.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((cos((a - b)) * (sin(b) * r)) / (cos((((b - a) - a) - b)) + cos((b - (a - (a + b)))))) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 70.8%

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

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -740000 \lor \neg \left(a \leq 6.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -740000.0) (not (<= a 6.5e-5)))
   (/ (* r (sin b)) (cos a))
   (* (/ r (cos b)) (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -740000.0) || !(a <= 6.5e-5)) {
		tmp = (r * sin(b)) / cos(a);
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-740000.0d0)) .or. (.not. (a <= 6.5d-5))) then
        tmp = (r * sin(b)) / cos(a)
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -740000.0) || !(a <= 6.5e-5)) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -740000.0) or not (a <= 6.5e-5):
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -740000.0) || !(a <= 6.5e-5))
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -740000.0) || ~((a <= 6.5e-5)))
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -740000.0], N[Not[LessEqual[a, 6.5e-5]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.4e5 or 6.49999999999999943e-5 < a

   1. Initial program 50.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 50.6

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

   5. Applied rewrites 50.6%

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

   if -7.4e5 < a < 6.49999999999999943e-5

   1. Initial program 96.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -740000 \lor \neg \left(a \leq 6.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000105 \lor \neg \left(b \leq 0.105\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.000105) (not (<= b 0.105)))
   (* (/ r (cos b)) (sin b))
   (/
    (*
     (fma
      (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
      (* b b)
      r)
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000105) || !(b <= 0.105)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.000105) || !(b <= 0.105))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.000105], N[Not[LessEqual[b, 0.105]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e-4 or 0.104999999999999996 < b

   1. Initial program 44.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 43.9

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

   5. Applied rewrites 43.9%

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

   if -1.05e-4 < b < 0.104999999999999996

   1. Initial program 98.5%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000105 \lor \neg \left(b \leq 0.105\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% 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 70.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 70.6

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

4. Applied rewrites 70.6%

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

Alternative 9: 76.7% 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 70.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 70.6

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

4. Applied rewrites 70.6%

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

Alternative 10: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 3.7\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.6) (not (<= b 3.7)))
   (/ (* r (sin b)) 1.0)
   (/
    (*
     (fma
      (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
      (* b b)
      r)
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.6) || !(b <= 3.7)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.6) || !(b <= 3.7))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.6], N[Not[LessEqual[b, 3.7]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5999999999999996 or 3.7000000000000002 < b

   1. Initial program 43.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 4.3%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, b, -a\right), \color{blue}{b}, 1\right)} \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 12.7%

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

   if -4.5999999999999996 < b < 3.7000000000000002

   1. Initial program 98.5%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 3.7\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9 \lor \neg \left(b \leq 3.6\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.9) (not (<= b 3.6)))
   (/ (* r (sin b)) 1.0)
   (/ (* b r) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.9) || !(b <= 3.6)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (b * r) / cos((a + b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-3.9d0)) .or. (.not. (b <= 3.6d0))) then
        tmp = (r * sin(b)) / 1.0d0
    else
        tmp = (b * r) / cos((a + b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.9) || !(b <= 3.6)) {
		tmp = (r * Math.sin(b)) / 1.0;
	} else {
		tmp = (b * r) / Math.cos((a + b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3.9) or not (b <= 3.6):
		tmp = (r * math.sin(b)) / 1.0
	else:
		tmp = (b * r) / math.cos((a + b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -3.9) || !(b <= 3.6))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -3.9) || ~((b <= 3.6)))
		tmp = (r * sin(b)) / 1.0;
	else
		tmp = (b * r) / cos((a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.9], N[Not[LessEqual[b, 3.6]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.89999999999999991 or 3.60000000000000009 < b

   1. Initial program 43.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 4.3%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, b, -a\right), \color{blue}{b}, 1\right)} \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 12.7%

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

   if -3.89999999999999991 < b < 3.60000000000000009

   1. Initial program 98.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \lor \neg \left(b \leq 3.6\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 1.6\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.6) (not (<= b 1.6)))
   (/ (* r (sin b)) 1.0)
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.6) || !(b <= 1.6)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.6d0)) .or. (.not. (b <= 1.6d0))) then
        tmp = (r * sin(b)) / 1.0d0
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.6) || !(b <= 1.6)) {
		tmp = (r * Math.sin(b)) / 1.0;
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.6) or not (b <= 1.6):
		tmp = (r * math.sin(b)) / 1.0
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.6) || !(b <= 1.6))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.6) || ~((b <= 1.6)))
		tmp = (r * sin(b)) / 1.0;
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.6], N[Not[LessEqual[b, 1.6]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5999999999999996 or 1.6000000000000001 < b

   1. Initial program 43.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 4.3%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, b, -a\right), \color{blue}{b}, 1\right)} \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 12.7%

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

   if -4.5999999999999996 < b < 1.6000000000000001

   1. Initial program 98.5%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 1.6\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. *-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 49.8

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

5. Applied rewrites 49.8%

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

7. Applied rewrites 49.8%

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

Alternative 14: 34.2% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. *-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 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 31.8%

   \[\leadsto b \cdot \color{blue}{r} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))