rsin A (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 11.6s

Alternatives: 13

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative 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-*.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 N/A

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

   15. 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. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

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

Alternative 4: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos \left(b - a\right) \cdot \frac{\sin b \cdot r}{0.5 \cdot \left(\cos \left(b - \left(a - \left(a + b\right)\right)\right) + \cos \left(\left(\left(b - a\right) - a\right) - b\right)\right)} \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return cos((b - a)) * ((sin(b) * r) / (0.5 * (cos((b - (a - (a + b)))) + cos((((b - a) - 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 = cos((b - a)) * ((sin(b) * r) / (0.5d0 * (cos((b - (a - (a + b)))) + cos((((b - a) - a) - b)))))
end function

Java

public static double code(double r, double a, double b) {
	return Math.cos((b - a)) * ((Math.sin(b) * r) / (0.5 * (Math.cos((b - (a - (a + b)))) + Math.cos((((b - a) - a) - b)))));
}

Python

def code(r, a, b):
	return math.cos((b - a)) * ((math.sin(b) * r) / (0.5 * (math.cos((b - (a - (a + b)))) + math.cos((((b - a) - a) - b)))))

Julia

function code(r, a, b)
	return Float64(cos(Float64(b - a)) * Float64(Float64(sin(b) * r) / Float64(0.5 * Float64(cos(Float64(b - Float64(a - Float64(a + b)))) + cos(Float64(Float64(Float64(b - a) - a) - b))))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = cos((b - a)) * ((sin(b) * r) / (0.5 * (cos((b - (a - (a + b)))) + cos((((b - a) - a) - b)))));
end

Wolfram

code[r_, a_, b_] := N[(N[Cos[N[(b - a), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(0.5 * N[(N[Cos[N[(b - N[(a - N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[(N[(b - a), $MachinePrecision] - a), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 73.2%

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

   1. lift-*.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-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

6. Applied rewrites 73.7%

   \[\leadsto \frac{\sin b \cdot r}{\color{blue}{\left(\cos \left(\left(\left(b - a\right) - a\right) - b\right) + \cos \left(b - \left(a - \left(a + b\right)\right)\right)\right) \cdot 0.5}} \cdot \cos \left(b - a\right) \]

7. Final simplification 73.7%

   \[\leadsto \cos \left(b - a\right) \cdot \frac{\sin b \cdot r}{0.5 \cdot \left(\cos \left(b - \left(a - \left(a + b\right)\right)\right) + \cos \left(\left(\left(b - a\right) - a\right) - b\right)\right)} \]
8. Add Preprocessing

Alternative 5: 76.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 73.2%

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

6. Applied rewrites 73.6%

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

7. Final simplification 73.6%

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

Alternative 6: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos b}\\ \mathbf{if}\;b \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 14000000000:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* (sin b) r) (cos b))))
   (if (<= b -4.4e-6) t_0 (if (<= b 14000000000.0) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) * r) / cos(b);
	double tmp;
	if (b <= -4.4e-6) {
		tmp = t_0;
	} else if (b <= 14000000000.0) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) * r) / cos(b)
    if (b <= (-4.4d-6)) then
        tmp = t_0
    else if (b <= 14000000000.0d0) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) * r) / Math.cos(b);
	double tmp;
	if (b <= -4.4e-6) {
		tmp = t_0;
	} else if (b <= 14000000000.0) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) * r) / math.cos(b)
	tmp = 0
	if b <= -4.4e-6:
		tmp = t_0
	elif b <= 14000000000.0:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) * r) / cos(b))
	tmp = 0.0
	if (b <= -4.4e-6)
		tmp = t_0;
	elseif (b <= 14000000000.0)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) * r) / cos(b);
	tmp = 0.0;
	if (b <= -4.4e-6)
		tmp = t_0;
	elseif (b <= 14000000000.0)
		tmp = (r / 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] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.4e-6], t$95$0, If[LessEqual[b, 14000000000.0], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.4000000000000002e-6 or 1.4e10 < b

   1. Initial program 50.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 50.5

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

   5. Applied rewrites 50.5%

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

   if -4.4000000000000002e-6 < b < 1.4e10

   1. Initial program 96.9%

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

   3. Taylor expanded in b around 0

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

      1. *-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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 14000000000:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 14000000000:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -4.4e-6) t_0 (if (<= b 14000000000.0) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -4.4e-6) {
		tmp = t_0;
	} else if (b <= 14000000000.0) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r / cos(b)) * sin(b)
    if (b <= (-4.4d-6)) then
        tmp = t_0
    else if (b <= 14000000000.0d0) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r / Math.cos(b)) * Math.sin(b);
	double tmp;
	if (b <= -4.4e-6) {
		tmp = t_0;
	} else if (b <= 14000000000.0) {
		tmp = (r / Math.cos(a)) * b;
	} 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 <= -4.4e-6:
		tmp = t_0
	elif b <= 14000000000.0:
		tmp = (r / math.cos(a)) * b
	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 <= -4.4e-6)
		tmp = t_0;
	elseif (b <= 14000000000.0)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (b <= -4.4e-6)
		tmp = t_0;
	elseif (b <= 14000000000.0)
		tmp = (r / cos(a)) * b;
	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, -4.4e-6], t$95$0, If[LessEqual[b, 14000000000.0], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.4000000000000002e-6 or 1.4e10 < b

   1. Initial program 50.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 50.5

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

   5. Applied rewrites 50.5%

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

   if -4.4000000000000002e-6 < b < 1.4e10

   1. Initial program 96.9%

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

   3. Taylor expanded in b around 0

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

      1. *-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 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 8: 76.5% 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 73.2%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 73.2

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

4. Applied rewrites 73.2%

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

Alternative 9: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{1}{\sin b}}\\ \mathbf{if}\;b \leq -45000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 22:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ 1.0 (sin b)))))
   (if (<= b -45000.0)
     t_0
     (if (<= b 22.0)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / (1.0 / sin(b));
	double tmp;
	if (b <= -45000.0) {
		tmp = t_0;
	} else if (b <= 22.0) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(1.0 / sin(b)))
	tmp = 0.0
	if (b <= -45000.0)
		tmp = t_0;
	elseif (b <= 22.0)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -45000.0], t$95$0, If[LessEqual[b, 22.0], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -45000 or 22 < b

   1. Initial program 48.4%

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

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. remove-double-div N/A

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

      2. unpow-1 N/A

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

      3. lift-pow.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-1 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 48.3

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 10.9

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

   9. Applied rewrites 10.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 12.0%

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

   if -45000 < b < 22

   1. Initial program 98.4%

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

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

   5. Applied rewrites 97.5%

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

Alternative 10: 54.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{1}{\sin b}}\\ \mathbf{if}\;b \leq -270000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.58:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ 1.0 (sin b)))))
   (if (<= b -270000000000.0) t_0 (if (<= b 0.58) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / (1.0 / sin(b));
	double tmp;
	if (b <= -270000000000.0) {
		tmp = t_0;
	} else if (b <= 0.58) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r / (1.0d0 / sin(b))
    if (b <= (-270000000000.0d0)) then
        tmp = t_0
    else if (b <= 0.58d0) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r / (1.0 / Math.sin(b));
	double tmp;
	if (b <= -270000000000.0) {
		tmp = t_0;
	} else if (b <= 0.58) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r / (1.0 / math.sin(b))
	tmp = 0
	if b <= -270000000000.0:
		tmp = t_0
	elif b <= 0.58:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r / Float64(1.0 / sin(b)))
	tmp = 0.0
	if (b <= -270000000000.0)
		tmp = t_0;
	elseif (b <= 0.58)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r / (1.0 / sin(b));
	tmp = 0.0;
	if (b <= -270000000000.0)
		tmp = t_0;
	elseif (b <= 0.58)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -270000000000.0], t$95$0, If[LessEqual[b, 0.58], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.7e11 or 0.57999999999999996 < b

   1. Initial program 48.4%

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

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. remove-double-div N/A

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

      2. unpow-1 N/A

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

      3. lift-pow.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-1 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 48.3

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 10.9

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

   9. Applied rewrites 10.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 12.0%

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

   if -2.7e11 < b < 0.57999999999999996

   1. Initial program 98.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 11: 50.8% 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 73.2%

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

3. Taylor expanded in b around 0

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

   1. *-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 50.3

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

5. Applied rewrites 50.3%

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

Alternative 12: 50.8% 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 73.2%

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

3. Taylor expanded in b around 0

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

   1. *-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 50.3

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

5. Applied rewrites 50.3%

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

7. Applied rewrites 50.2%

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

Alternative 13: 34.6% 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 73.2%

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

3. Taylor expanded in b around 0

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

   1. *-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 50.3

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

5. Applied rewrites 50.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.7%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))