rsin B (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 13.6s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-sin.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. sin-neg N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. mul-1-neg N/A

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

   11. sin-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-cos.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-sin.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. cos-sum N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.000106)
   (* (sin b) (/ r (cos b)))
   (if (<= b 98.0) (/ (* r (sin b)) (cos a)) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.000106) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 98.0) {
		tmp = (r * sin(b)) / cos(a);
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.000106d0)) then
        tmp = sin(b) * (r / cos(b))
    else if (b <= 98.0d0) then
        tmp = (r * sin(b)) / cos(a)
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.000106) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (b <= 98.0) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.000106:
		tmp = math.sin(b) * (r / math.cos(b))
	elif b <= 98.0:
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.000106)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 98.0)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.000106)
		tmp = sin(b) * (r / cos(b));
	elseif (b <= 98.0)
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.000106], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 98.0], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.06e-4

   1. Initial program 61.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 61.5

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if -1.06e-4 < b < 98

   1. Initial program 98.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 98.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 98.3

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

   7. Applied rewrites 98.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if 98 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 50.2

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

   5. Applied rewrites 50.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

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

      9. lower-tan.f64 50.4

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

   7. Applied rewrites 50.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.000106)
   (* (sin b) (/ r (cos b)))
   (if (<= b 98.0) (* (sin b) (/ r (cos a))) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.000106) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 98.0) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.000106d0)) then
        tmp = sin(b) * (r / cos(b))
    else if (b <= 98.0d0) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.000106) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (b <= 98.0) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.000106:
		tmp = math.sin(b) * (r / math.cos(b))
	elif b <= 98.0:
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.000106)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 98.0)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.000106)
		tmp = sin(b) * (r / cos(b));
	elseif (b <= 98.0)
		tmp = sin(b) * (r / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.000106], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 98.0], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.06e-4

   1. Initial program 61.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 61.5

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if -1.06e-4 < b < 98

   1. Initial program 98.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 98.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 98 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 50.2

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

   5. Applied rewrites 50.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

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

      9. lower-tan.f64 50.4

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

   7. Applied rewrites 50.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.000106) t_0 (if (<= b 98.0) (* (sin b) (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.000106) {
		tmp = t_0;
	} else if (b <= 98.0) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-0.000106d0)) then
        tmp = t_0
    else if (b <= 98.0d0) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -0.000106) {
		tmp = t_0;
	} else if (b <= 98.0) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -0.000106:
		tmp = t_0
	elif b <= 98.0:
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -0.000106)
		tmp = t_0;
	elseif (b <= 98.0)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -0.000106)
		tmp = t_0;
	elseif (b <= 98.0)
		tmp = sin(b) * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.000106], t$95$0, If[LessEqual[b, 98.0], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.06e-4 or 98 < b

   1. Initial program 55.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

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

      9. lower-tan.f64 55.8

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

   7. Applied rewrites 55.8%

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

   if -1.06e-4 < b < 98

   1. Initial program 98.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 98.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 98.3

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

   7. Applied rewrites 98.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 77.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 77.9

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

4. Applied rewrites 77.9%

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

5. Final simplification 77.9%

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

Alternative 9: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Final simplification 77.9%

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

Alternative 10: 76.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.000106) t_0 (if (<= b 98.0) (/ (* r b) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.000106) {
		tmp = t_0;
	} else if (b <= 98.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-0.000106d0)) then
        tmp = t_0
    else if (b <= 98.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -0.000106) {
		tmp = t_0;
	} else if (b <= 98.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -0.000106:
		tmp = t_0
	elif b <= 98.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -0.000106)
		tmp = t_0;
	elseif (b <= 98.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -0.000106)
		tmp = t_0;
	elseif (b <= 98.0)
		tmp = (r * b) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.000106], t$95$0, If[LessEqual[b, 98.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.06e-4 or 98 < b

   1. Initial program 55.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

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

      9. lower-tan.f64 55.8

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

   7. Applied rewrites 55.8%

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

   if -1.06e-4 < b < 98

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-cos.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.000106) t_0 (if (<= b 98.0) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.000106) {
		tmp = t_0;
	} else if (b <= 98.0) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-0.000106d0)) then
        tmp = t_0
    else if (b <= 98.0d0) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -0.000106) {
		tmp = t_0;
	} else if (b <= 98.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -0.000106:
		tmp = t_0
	elif b <= 98.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -0.000106)
		tmp = t_0;
	elseif (b <= 98.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -0.000106)
		tmp = t_0;
	elseif (b <= 98.0)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.000106], t$95$0, If[LessEqual[b, 98.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.06e-4 or 98 < b

   1. Initial program 55.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. quot-tan N/A

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

      9. lower-tan.f64 55.8

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

   7. Applied rewrites 55.8%

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

   if -1.06e-4 < b < 98

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000106:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 62.0

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

5. Applied rewrites 62.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. quot-tan N/A

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

   9. lower-tan.f64 62.1

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

7. Applied rewrites 62.1%

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

8. Final simplification 62.1%

   \[\leadsto r \cdot \tan b \]
9. Add Preprocessing

Alternative 13: 38.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 77.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 77.9

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

4. Applied rewrites 77.9%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 57.0

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

7. Applied rewrites 57.0%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.6

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

10. Applied rewrites 41.6%

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

Alternative 14: 34.2% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 37.3

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

8. Applied rewrites 37.3%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))