rsin A (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 12.1s

Alternatives: 19

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: 76.5% 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

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / fma(cos(b), cos(a), Float64(sin(a) * Float64(-sin(b)))))
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[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

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

Alternative 2: 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 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. unsub-neg N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-rgt-neg-out N/A

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

   13. lift-neg.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. distribute-rgt-neg-out N/A

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

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

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

   18. lift-neg.f64 N/A

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

   19. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 76.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (cos((a - b)) / 0.5d0) * ((sin(b) * r) / (cos((b - (a - (a + b)))) + cos((((b - a) - b) - a))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

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

6. Applied rewrites 75.7%

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos \left(\left(\left(b - a\right) - a\right) - b\right) + \cos \left(b - \left(a - \left(a + b\right)\right)\right)} \cdot \frac{\cos \left(a - b\right)}{0.5}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 76.1

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

8. Applied rewrites 76.1%

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

9. Final simplification 76.1%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= a -0.000145)
     (/ t_0 (cos a))
     (if (<= a 6.2e-5) (/ t_0 (cos b)) (* (/ r (cos a)) (sin b))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (a <= -0.000145) {
		tmp = t_0 / cos(a);
	} else if (a <= 6.2e-5) {
		tmp = t_0 / cos(b);
	} else {
		tmp = (r / cos(a)) * 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) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * r
    if (a <= (-0.000145d0)) then
        tmp = t_0 / cos(a)
    else if (a <= 6.2d-5) then
        tmp = t_0 / cos(b)
    else
        tmp = (r / cos(a)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * r;
	double tmp;
	if (a <= -0.000145) {
		tmp = t_0 / Math.cos(a);
	} else if (a <= 6.2e-5) {
		tmp = t_0 / Math.cos(b);
	} else {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if a <= -0.000145:
		tmp = t_0 / math.cos(a)
	elif a <= 6.2e-5:
		tmp = t_0 / math.cos(b)
	else:
		tmp = (r / math.cos(a)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (a <= -0.000145)
		tmp = Float64(t_0 / cos(a));
	elseif (a <= 6.2e-5)
		tmp = Float64(t_0 / cos(b));
	else
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * r;
	tmp = 0.0;
	if (a <= -0.000145)
		tmp = t_0 / cos(a);
	elseif (a <= 6.2e-5)
		tmp = t_0 / cos(b);
	else
		tmp = (r / cos(a)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -0.000145], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.2e-5], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e-4

   1. Initial program 45.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.9

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

   5. Applied rewrites 47.9%

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

   if -1.45e-4 < a < 6.20000000000000027e-5

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 6.20000000000000027e-5 < a

   1. Initial program 61.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. 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 61.6

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 58.9

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

   7. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.9

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

   9. Applied rewrites 58.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000145)
   (/ (* (sin b) r) (cos a))
   (if (<= a 6.2e-5) (* (/ (sin b) (cos b)) r) (* (/ r (cos a)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000145) {
		tmp = (sin(b) * r) / cos(a);
	} else if (a <= 6.2e-5) {
		tmp = (sin(b) / cos(b)) * r;
	} else {
		tmp = (r / cos(a)) * 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 <= (-0.000145d0)) then
        tmp = (sin(b) * r) / cos(a)
    else if (a <= 6.2d-5) then
        tmp = (sin(b) / cos(b)) * r
    else
        tmp = (r / cos(a)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000145) {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	} else if (a <= 6.2e-5) {
		tmp = (Math.sin(b) / Math.cos(b)) * r;
	} else {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000145:
		tmp = (math.sin(b) * r) / math.cos(a)
	elif a <= 6.2e-5:
		tmp = (math.sin(b) / math.cos(b)) * r
	else:
		tmp = (r / math.cos(a)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000145)
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	elseif (a <= 6.2e-5)
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	else
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000145)
		tmp = (sin(b) * r) / cos(a);
	elseif (a <= 6.2e-5)
		tmp = (sin(b) / cos(b)) * r;
	else
		tmp = (r / cos(a)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000145], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.2e-5], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e-4

   1. Initial program 45.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.9

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

   5. Applied rewrites 47.9%

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

   if -1.45e-4 < a < 6.20000000000000027e-5

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 6.20000000000000027e-5 < a

   1. Initial program 61.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. 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 61.6

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 58.9

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

   7. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.9

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

   9. Applied rewrites 58.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000145)
   (* (/ (sin b) (cos a)) r)
   (if (<= a 6.2e-5) (* (/ (sin b) (cos b)) r) (* (/ r (cos a)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000145) {
		tmp = (sin(b) / cos(a)) * r;
	} else if (a <= 6.2e-5) {
		tmp = (sin(b) / cos(b)) * r;
	} else {
		tmp = (r / cos(a)) * 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 <= (-0.000145d0)) then
        tmp = (sin(b) / cos(a)) * r
    else if (a <= 6.2d-5) then
        tmp = (sin(b) / cos(b)) * r
    else
        tmp = (r / cos(a)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000145) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else if (a <= 6.2e-5) {
		tmp = (Math.sin(b) / Math.cos(b)) * r;
	} else {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000145:
		tmp = (math.sin(b) / math.cos(a)) * r
	elif a <= 6.2e-5:
		tmp = (math.sin(b) / math.cos(b)) * r
	else:
		tmp = (r / math.cos(a)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000145)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	elseif (a <= 6.2e-5)
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	else
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000145)
		tmp = (sin(b) / cos(a)) * r;
	elseif (a <= 6.2e-5)
		tmp = (sin(b) / cos(b)) * r;
	else
		tmp = (r / cos(a)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000145], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[a, 6.2e-5], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e-4

   1. Initial program 45.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 45.1

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.9

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

   7. Applied rewrites 47.9%

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

   if -1.45e-4 < a < 6.20000000000000027e-5

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 6.20000000000000027e-5 < a

   1. Initial program 61.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. 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 61.6

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 58.9

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

   7. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.9

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

   9. Applied rewrites 58.9%

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

Alternative 8: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000145)
   (* (/ (sin b) (cos a)) r)
   (if (<= a 6.2e-5) (* (/ r (cos b)) (sin b)) (* (/ r (cos a)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000145) {
		tmp = (sin(b) / cos(a)) * r;
	} else if (a <= 6.2e-5) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (r / cos(a)) * 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 <= (-0.000145d0)) then
        tmp = (sin(b) / cos(a)) * r
    else if (a <= 6.2d-5) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (r / cos(a)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000145) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else if (a <= 6.2e-5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000145:
		tmp = (math.sin(b) / math.cos(a)) * r
	elif a <= 6.2e-5:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (r / math.cos(a)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000145)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	elseif (a <= 6.2e-5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000145)
		tmp = (sin(b) / cos(a)) * r;
	elseif (a <= 6.2e-5)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (r / cos(a)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000145], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[a, 6.2e-5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e-4

   1. Initial program 45.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 45.1

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 47.9

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

   7. Applied rewrites 47.9%

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

   if -1.45e-4 < a < 6.20000000000000027e-5

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 6.20000000000000027e-5 < a

   1. Initial program 61.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. 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 61.6

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 58.9

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

   7. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.9

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

   9. Applied rewrites 58.9%

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

Alternative 9: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos a} \cdot \sin b\\ \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos a)) (sin b))))
   (if (<= a -0.000145) t_0 (if (<= a 6.2e-5) (* (/ r (cos b)) (sin b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(a)) * sin(b);
	double tmp;
	if (a <= -0.000145) {
		tmp = t_0;
	} else if (a <= 6.2e-5) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r / cos(a)) * sin(b)
    if (a <= (-0.000145d0)) then
        tmp = t_0
    else if (a <= 6.2d-5) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r / Math.cos(a)) * Math.sin(b);
	double tmp;
	if (a <= -0.000145) {
		tmp = t_0;
	} else if (a <= 6.2e-5) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(a)) * math.sin(b)
	tmp = 0
	if a <= -0.000145:
		tmp = t_0
	elif a <= 6.2e-5:
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(a)) * sin(b))
	tmp = 0.0
	if (a <= -0.000145)
		tmp = t_0;
	elseif (a <= 6.2e-5)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(a)) * sin(b);
	tmp = 0.0;
	if (a <= -0.000145)
		tmp = t_0;
	elseif (a <= 6.2e-5)
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.000145], t$95$0, If[LessEqual[a, 6.2e-5], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e-4 or 6.20000000000000027e-5 < a

   1. Initial program 54.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 55.0

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 54.5

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

   7. Applied rewrites 54.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 54.4

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

   9. Applied rewrites 54.4%

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

   if -1.45e-4 < a < 6.20000000000000027e-5

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 10: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -310000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot r\right) \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -310000000000.0)
     t_0
     (if (<= b 2.7e-17)
       (/ (* (* (fma (* b b) -0.16666666666666666 1.0) r) b) (cos a))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -310000000000.0) {
		tmp = t_0;
	} else if (b <= 2.7e-17) {
		tmp = ((fma((b * b), -0.16666666666666666, 1.0) * r) * b) / cos(a);
	} 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 <= -310000000000.0)
		tmp = t_0;
	elseif (b <= 2.7e-17)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * r) * b) / cos(a));
	else
		tmp = t_0;
	end
	return 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, -310000000000.0], t$95$0, If[LessEqual[b, 2.7e-17], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.1e11 or 2.7000000000000001e-17 < b

   1. Initial program 56.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 56.0

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

   5. Applied rewrites 56.0%

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

   if -3.1e11 < b < 2.7000000000000001e-17

   1. Initial program 97.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in b around 0

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

      1. lower-cos.f64 97.7

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

   10. Applied rewrites 97.7%

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

Alternative 11: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Final simplification 75.8%

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

Alternative 12: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.8

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

4. Applied rewrites 75.8%

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

Alternative 13: 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 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.8

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

4. Applied rewrites 75.8%

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

Alternative 14: 54.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{1}{\sin b}}\\ \mathbf{if}\;b \leq -3200000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot r\right) \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ 1.0 (sin b)))))
   (if (<= b -3200000000000.0)
     t_0
     (if (<= b 2.7e-17)
       (/ (* (* (fma (* b b) -0.16666666666666666 1.0) r) b) (cos a))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r / (1.0 / sin(b));
	double tmp;
	if (b <= -3200000000000.0) {
		tmp = t_0;
	} else if (b <= 2.7e-17) {
		tmp = ((fma((b * b), -0.16666666666666666, 1.0) * r) * b) / cos(a);
	} 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 <= -3200000000000.0)
		tmp = t_0;
	elseif (b <= 2.7e-17)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * r) * b) / cos(a));
	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, -3200000000000.0], t$95$0, If[LessEqual[b, 2.7e-17], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e12 or 2.7000000000000001e-17 < b

   1. Initial program 56.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 56.4

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 12.2%

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

   if -3.2e12 < b < 2.7000000000000001e-17

   1. Initial program 97.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in b around 0

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

      1. lower-cos.f64 97.7

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

   10. Applied rewrites 97.7%

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

Alternative 15: 54.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\frac{1}{\sin b}}\\ \mathbf{if}\;b \leq -320000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ r (/ 1.0 (sin b)))))
   (if (<= b -320000000000.0)
     t_0
     (if (<= b 9.8e+14) (/ (* b 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 <= -320000000000.0) {
		tmp = t_0;
	} else if (b <= 9.8e+14) {
		tmp = (b * 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 <= (-320000000000.0d0)) then
        tmp = t_0
    else if (b <= 9.8d+14) then
        tmp = (b * 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 <= -320000000000.0) {
		tmp = t_0;
	} else if (b <= 9.8e+14) {
		tmp = (b * 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 <= -320000000000.0:
		tmp = t_0
	elif b <= 9.8e+14:
		tmp = (b * 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 <= -320000000000.0)
		tmp = t_0;
	elseif (b <= 9.8e+14)
		tmp = Float64(Float64(b * r) / cos(Float64(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 <= -320000000000.0)
		tmp = t_0;
	elseif (b <= 9.8e+14)
		tmp = (b * 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, -320000000000.0], t$95$0, If[LessEqual[b, 9.8e+14], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e11 or 9.8e14 < b

   1. Initial program 55.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.6%

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

   if -3.2e11 < b < 9.8e14

   1. Initial program 97.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 16: 50.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 48.1

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

5. Applied rewrites 48.1%

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

Alternative 17: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 47.9

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

5. Applied rewrites 47.9%

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

7. Applied rewrites 47.9%

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

Alternative 18: 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 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 47.9

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

5. Applied rewrites 47.9%

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

7. Applied rewrites 47.9%

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

8. Final simplification 47.9%

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

Alternative 19: 35.0% 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 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 47.9

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

5. Applied rewrites 47.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 31.4%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))