rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 10.6s

Alternatives: 13

Speedup: 0.5×

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: 76.2% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. +-lft-identity N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{0 + \sin b}, -\sin a, \cos b \cdot \cos a\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\frac{0 \cdot 0 - \sin b \cdot \sin b}{0 - \sin b}}, -\sin a, \cos b \cdot \cos a\right)} \]

   3. neg-sub0 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\frac{0 \cdot 0 - \sin b \cdot \sin b}{\color{blue}{\mathsf{neg}\left(\sin b\right)}}, -\sin a, \cos b \cdot \cos a\right)} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\frac{0 \cdot 0 - \sin b \cdot \sin b}{\color{blue}{-\sin b}}, -\sin a, \cos b \cdot \cos a\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\color{blue}{\frac{0 \cdot 0 - \sin b \cdot \sin b}{-\sin b}}, -\sin a, \cos b \cdot \cos a\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\frac{\color{blue}{0} - \sin b \cdot \sin b}{-\sin b}, -\sin a, \cos b \cdot \cos a\right)} \]

   7. sub0-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. unsub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. cos-sum N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. associate-/l* N/A

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

6. Applied rewrites 79.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. div-inv N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. cos-sum N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. div-sub N/A

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

   14. sub-neg N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. *-lft-identity N/A

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

   18. times-frac N/A

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

   19. /-rgt-identity N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. unsub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. cos-sum N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. associate-/l* N/A

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

6. Applied rewrites 79.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. div-inv N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. cos-sum N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. div-sub N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-sin.f64 N/A

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

   20. lift-sin.f64 N/A

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

   21. *-lft-identity N/A

       \[\leadsto \frac{r}{\frac{\cos a \cdot \cos b}{\sin b} - \frac{\sin a \cdot \sin b}{\color{blue}{1 \cdot \sin b}}} \]

8. Applied rewrites 99.5%

   \[\leadsto \frac{r}{\color{blue}{\frac{\cos a \cdot \cos b}{\sin b} - \frac{\sin a}{1} \cdot 1}} \]

9. Final simplification 99.5%

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

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -0.0275:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.045:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -0.0275)
     t_0
     (if (<= b 0.045)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -0.0275) {
		tmp = t_0;
	} else if (b <= 0.045) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -0.0275)
		tmp = t_0;
	elseif (b <= 0.045)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0275], t$95$0, If[LessEqual[b, 0.045], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0275000000000000001 or 0.044999999999999998 < b

   1. Initial program 55.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 55.6

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

   5. Applied rewrites 55.6%

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

   if -0.0275000000000000001 < b < 0.044999999999999998

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

Alternative 7: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-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 79.4

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

4. Applied rewrites 79.4%

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

Alternative 8: 53.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b} \cdot \cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 5e+27)
   (/ r (* (/ (fma (* b b) 0.16666666666666666 1.0) b) (cos (+ a b))))
   (/ (* (sin b) r) 1.0)))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 5e+27) {
		tmp = r / ((fma((b * b), 0.16666666666666666, 1.0) / b) * cos((a + b)));
	} else {
		tmp = (sin(b) * r) / 1.0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 5e+27)
		tmp = Float64(r / Float64(Float64(fma(Float64(b * b), 0.16666666666666666, 1.0) / b) * cos(Float64(a + b))));
	else
		tmp = Float64(Float64(sin(b) * r) / 1.0);
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 5e+27], N[(r / N[(N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / b), $MachinePrecision] * N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999979e27

   1. Initial program 86.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. unsub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. associate-/l* N/A

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

   6. Applied rewrites 86.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.0

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

   9. Applied rewrites 71.0%

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

   if 4.99999999999999979e27 < b

   1. Initial program 54.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 12.1%

       \[\leadsto \frac{r \cdot \sin b}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b} \cdot \cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+28}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 1.12e+28) (/ (* b r) (cos a)) (/ (* (sin b) r) 1.0)))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 1.12e+28) {
		tmp = (b * r) / cos(a);
	} else {
		tmp = (sin(b) * r) / 1.0;
	}
	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) :: tmp
    if (b <= 1.12d+28) then
        tmp = (b * r) / cos(a)
    else
        tmp = (sin(b) * r) / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 1.12e+28) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = (Math.sin(b) * r) / 1.0;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 1.12e+28:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = (math.sin(b) * r) / 1.0
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 1.12e+28)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = Float64(Float64(sin(b) * r) / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 1.12e+28)
		tmp = (b * r) / cos(a);
	else
		tmp = (sin(b) * r) / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 1.12e+28], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.12e28

   1. Initial program 86.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 71.0

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 71.0%

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

   if 1.12e28 < b

   1. Initial program 54.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 12.1%

       \[\leadsto \frac{r \cdot \sin b}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+28}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 55.9%

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

Alternative 11: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 55.9

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

5. Applied rewrites 55.9%

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

Alternative 12: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 55.9%

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

8. Final simplification 55.9%

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

Alternative 13: 33.9% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 55.9

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

5. Applied rewrites 55.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.3%

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

Reproduce

herbie shell --seed 2024327 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))