rsin A (should all be same)

Percentage Accurate: 76.4% → 99.5%

Time: 11.9s

Alternatives: 9

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.3

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

4. Applied rewrites 78.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

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

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

   11. lift-sin.f64 N/A

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

   12. sin-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

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(sin(b) * Float64(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[Sin[b], $MachinePrecision] * N[(r / N[(N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.3

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

4. Applied rewrites 78.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.2% accurate, 0.4× speedup

Math

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

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(sin(b) * Float64(r / fma(cos(b), cos(a), Float64(sin(a) * sin(b)))))
end

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.3

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

4. Applied rewrites 78.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-div N/A

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

   7. lift-/.f64 N/A

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

   8. inv-pow N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. sqr-neg N/A

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

   12. neg-mul-1 N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. lift-/.f64 N/A

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

   16. neg-mul-1 N/A

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

   17. lift-/.f64 N/A

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

   18. div-inv N/A

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

   19. lift-/.f64 N/A

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

   20. pow-prod-down N/A

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

8. Applied rewrites 78.4%

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

9. Final simplification 78.4%

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

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -6.5e-5)
   (/ (* r (sin b)) (cos b))
   (if (<= b 2e-20) (* b (/ r (cos a))) (* (sin b) (/ r (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.5e-5) {
		tmp = (r * sin(b)) / cos(b);
	} else if (b <= 2e-20) {
		tmp = b * (r / cos(a));
	} else {
		tmp = sin(b) * (r / cos(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 <= (-6.5d-5)) then
        tmp = (r * sin(b)) / cos(b)
    else if (b <= 2d-20) then
        tmp = b * (r / cos(a))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.5e-5) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else if (b <= 2e-20) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -6.5e-5:
		tmp = (r * math.sin(b)) / math.cos(b)
	elif b <= 2e-20:
		tmp = b * (r / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -6.5e-5)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	elseif (b <= 2e-20)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -6.5e-5)
		tmp = (r * sin(b)) / cos(b);
	elseif (b <= 2e-20)
		tmp = b * (r / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -6.5e-5], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-20], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.49999999999999943e-5

   1. Initial program 61.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

   if -6.49999999999999943e-5 < b < 1.99999999999999989e-20

   1. Initial program 99.7%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   if 1.99999999999999989e-20 < b

   1. Initial program 60.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 61.3

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

   5. Applied rewrites 61.3%

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

Alternative 5: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos b)))))
   (if (<= b -6.5e-5) t_0 (if (<= b 2e-20) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(b));
	double tmp;
	if (b <= -6.5e-5) {
		tmp = t_0;
	} else if (b <= 2e-20) {
		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 = sin(b) * (r / cos(b))
    if (b <= (-6.5d-5)) then
        tmp = t_0
    else if (b <= 2d-20) 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 = Math.sin(b) * (r / Math.cos(b));
	double tmp;
	if (b <= -6.5e-5) {
		tmp = t_0;
	} else if (b <= 2e-20) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(b))
	tmp = 0
	if b <= -6.5e-5:
		tmp = t_0
	elif b <= 2e-20:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(b)))
	tmp = 0.0
	if (b <= -6.5e-5)
		tmp = t_0;
	elseif (b <= 2e-20)
		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 = sin(b) * (r / cos(b));
	tmp = 0.0;
	if (b <= -6.5e-5)
		tmp = t_0;
	elseif (b <= 2e-20)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.5e-5], t$95$0, If[LessEqual[b, 2e-20], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.49999999999999943e-5 or 1.99999999999999989e-20 < b

   1. Initial program 60.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if -6.49999999999999943e-5 < b < 1.99999999999999989e-20

   1. Initial program 99.7%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

Alternative 6: 76.4% 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 78.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.3

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

4. Applied rewrites 78.3%

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

5. Final simplification 78.3%

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

Alternative 7: 54.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5500000000000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -5.8e+20)
     t_0
     (if (<= b 5500000000000.0) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -5.8e+20) {
		tmp = t_0;
	} else if (b <= 5500000000000.0) {
		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 = sin(b) * (r / 1.0d0)
    if (b <= (-5.8d+20)) then
        tmp = t_0
    else if (b <= 5500000000000.0d0) 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 = Math.sin(b) * (r / 1.0);
	double tmp;
	if (b <= -5.8e+20) {
		tmp = t_0;
	} else if (b <= 5500000000000.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / 1.0)
	tmp = 0
	if b <= -5.8e+20:
		tmp = t_0
	elif b <= 5500000000000.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -5.8e+20)
		tmp = t_0;
	elseif (b <= 5500000000000.0)
		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 = sin(b) * (r / 1.0);
	tmp = 0.0;
	if (b <= -5.8e+20)
		tmp = t_0;
	elseif (b <= 5500000000000.0)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.8e+20], t$95$0, If[LessEqual[b, 5500000000000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.8e20 or 5.5e12 < b

   1. Initial program 58.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 58.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 10.4

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

   7. Applied rewrites 10.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.0%

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

   if -5.8e20 < b < 5.5e12

   1. Initial program 99.7%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 96.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+20}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 5500000000000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 1.9× speedup

Math

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

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(b * Float64(r / cos(a)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 47.8

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

5. Applied rewrites 47.8%

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

7. Applied rewrites 47.9%

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

Alternative 9: 34.5% 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 78.2%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 47.8

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

5. Applied rewrites 47.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 34.4%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))