rsin A (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 11.0s

Alternatives: 12

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: 76.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

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

   2. unpow-1 N/A

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

   3. lift-pow.f64 N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow-1 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

8. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

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

   2. unpow-1 N/A

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

   3. lift-pow.f64 N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow-1 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.5

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

   7. lift-fma.f64 N/A

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

8. Applied rewrites 99.5%

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

Alternative 4: 76.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return ((sin(b) * r) / ((cos((((b - a) - b) - a)) + cos((b - (a - (a + b))))) * 0.5)) * 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) - b) - a)) + cos((b - (a - (a + b))))) * 0.5d0)) * 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) - b) - a)) + Math.cos((b - (a - (a + b))))) * 0.5)) * Math.cos((b - a));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. cos-sum N/A

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

   5. flip-- N/A

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

   6. cos-diff N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 75.4%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 75.3%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 75.8

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

8. Applied rewrites 75.8%

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

Alternative 5: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b \cdot r}{\left(\cos \left(-2 \cdot a\right) + \cos \left(b - \left(a - \left(a + b\right)\right)\right)\right) \cdot 0.5} \cdot \cos \left(b - a\right) \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (*
  (/ (* (sin b) r) (* (+ (cos (* -2.0 a)) (cos (- b (- a (+ a b))))) 0.5))
  (cos (- b a))))

C

double code(double r, double a, double b) {
	return ((sin(b) * r) / ((cos((-2.0 * a)) + cos((b - (a - (a + b))))) * 0.5)) * 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(((-2.0d0) * a)) + cos((b - (a - (a + b))))) * 0.5d0)) * cos((b - a))
end function

Java

public static double code(double r, double a, double b) {
	return ((Math.sin(b) * r) / ((Math.cos((-2.0 * a)) + Math.cos((b - (a - (a + b))))) * 0.5)) * Math.cos((b - a));
}

Python

def code(r, a, b):
	return ((math.sin(b) * r) / ((math.cos((-2.0 * a)) + math.cos((b - (a - (a + b))))) * 0.5)) * math.cos((b - a))

Julia

function code(r, a, b)
	return Float64(Float64(Float64(sin(b) * r) / Float64(Float64(cos(Float64(-2.0 * a)) + cos(Float64(b - Float64(a - Float64(a + b))))) * 0.5)) * cos(Float64(b - a)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = ((sin(b) * r) / ((cos((-2.0 * a)) + cos((b - (a - (a + b))))) * 0.5)) * cos((b - a));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[(N[Cos[N[(-2.0 * a), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(b - N[(a - N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. cos-sum N/A

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

   5. flip-- N/A

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

   6. cos-diff N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 75.4%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 75.3%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 75.6

      \[\leadsto \frac{\sin b \cdot r}{\left(\cos \color{blue}{\left(-2 \cdot a\right)} + \cos \left(b - \left(a - \left(a + b\right)\right)\right)\right) \cdot 0.5} \cdot \cos \left(b - a\right) \]

9. Applied rewrites 75.6%

   \[\leadsto \frac{\sin b \cdot r}{\left(\cos \color{blue}{\left(-2 \cdot a\right)} + \cos \left(b - \left(a - \left(a + b\right)\right)\right)\right) \cdot 0.5} \cdot \cos \left(b - a\right) \]
10. Add Preprocessing

Alternative 6: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -280000:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-8}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -280000.0)
   (* (tan b) r)
   (if (<= b 1.08e-8) (/ (* b r) (cos (+ a b))) (* (/ r (cos b)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -280000.0) {
		tmp = tan(b) * r;
	} else if (b <= 1.08e-8) {
		tmp = (b * r) / cos((a + b));
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-280000.0d0)) then
        tmp = tan(b) * r
    else if (b <= 1.08d-8) then
        tmp = (b * r) / cos((a + b))
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -280000.0) {
		tmp = Math.tan(b) * r;
	} else if (b <= 1.08e-8) {
		tmp = (b * r) / Math.cos((a + b));
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -280000.0:
		tmp = math.tan(b) * r
	elif b <= 1.08e-8:
		tmp = (b * r) / math.cos((a + b))
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -280000.0)
		tmp = Float64(tan(b) * r);
	elseif (b <= 1.08e-8)
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -280000.0)
		tmp = tan(b) * r;
	elseif (b <= 1.08e-8)
		tmp = (b * r) / cos((a + b));
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -280000.0], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], If[LessEqual[b, 1.08e-8], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8e5

   1. Initial program 56.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.8

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

   7. Applied rewrites 54.8%

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

   9. Applied rewrites 54.8%

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

   if -2.8e5 < b < 1.0800000000000001e-8

   1. Initial program 96.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1.0800000000000001e-8 < b

   1. Initial program 53.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 53.3

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

   5. Applied rewrites 53.3%

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

Alternative 7: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 75.5

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

4. Applied rewrites 75.5%

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

Alternative 8: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -280000 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -280000.0) (not (<= b 1.08e-8)))
   (* (tan b) r)
   (/ (* b r) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -280000.0) || !(b <= 1.08e-8)) {
		tmp = tan(b) * r;
	} else {
		tmp = (b * r) / cos((a + 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 <= (-280000.0d0)) .or. (.not. (b <= 1.08d-8))) then
        tmp = tan(b) * r
    else
        tmp = (b * r) / cos((a + b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -280000.0) || !(b <= 1.08e-8)) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = (b * r) / Math.cos((a + b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -280000.0) or not (b <= 1.08e-8):
		tmp = math.tan(b) * r
	else:
		tmp = (b * r) / math.cos((a + b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -280000.0) || !(b <= 1.08e-8))
		tmp = Float64(tan(b) * r);
	else
		tmp = Float64(Float64(b * r) / cos(Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -280000.0) || ~((b <= 1.08e-8)))
		tmp = tan(b) * r;
	else
		tmp = (b * r) / cos((a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -280000.0], N[Not[LessEqual[b, 1.08e-8]], $MachinePrecision]], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8e5 or 1.0800000000000001e-8 < b

   1. Initial program 54.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 53.9

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

   7. Applied rewrites 53.9%

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

   9. Applied rewrites 54.0%

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

   if -2.8e5 < b < 1.0800000000000001e-8

   1. Initial program 96.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -280000 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -280000 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -280000.0) (not (<= b 1.08e-8)))
   (* (tan b) r)
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -280000.0) || !(b <= 1.08e-8)) {
		tmp = tan(b) * r;
	} else {
		tmp = (b * r) / cos(a);
	}
	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 <= (-280000.0d0)) .or. (.not. (b <= 1.08d-8))) then
        tmp = tan(b) * r
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -280000.0) || !(b <= 1.08e-8)) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -280000.0) or not (b <= 1.08e-8):
		tmp = math.tan(b) * r
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -280000.0) || !(b <= 1.08e-8))
		tmp = Float64(tan(b) * r);
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -280000.0) || ~((b <= 1.08e-8)))
		tmp = tan(b) * r;
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -280000.0], N[Not[LessEqual[b, 1.08e-8]], $MachinePrecision]], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8e5 or 1.0800000000000001e-8 < b

   1. Initial program 54.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 53.9

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

   7. Applied rewrites 53.9%

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

   9. Applied rewrites 54.0%

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

   if -2.8e5 < b < 1.0800000000000001e-8

   1. Initial program 96.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. *-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 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -280000 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -280000 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -280000.0) (not (<= b 1.08e-8)))
   (* (tan b) r)
   (* (/ r (cos a)) b)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -280000.0) || !(b <= 1.08e-8)) {
		tmp = tan(b) * r;
	} else {
		tmp = (r / cos(a)) * 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 <= (-280000.0d0)) .or. (.not. (b <= 1.08d-8))) then
        tmp = tan(b) * r
    else
        tmp = (r / cos(a)) * b
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -280000.0) || !(b <= 1.08e-8)) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = (r / Math.cos(a)) * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -280000.0) or not (b <= 1.08e-8):
		tmp = math.tan(b) * r
	else:
		tmp = (r / math.cos(a)) * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -280000.0) || !(b <= 1.08e-8))
		tmp = Float64(tan(b) * r);
	else
		tmp = Float64(Float64(r / cos(a)) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -280000.0) || ~((b <= 1.08e-8)))
		tmp = tan(b) * r;
	else
		tmp = (r / cos(a)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -280000.0], N[Not[LessEqual[b, 1.08e-8]], $MachinePrecision]], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8e5 or 1.0800000000000001e-8 < b

   1. Initial program 54.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 53.9

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

   7. Applied rewrites 53.9%

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

   9. Applied rewrites 54.0%

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

   if -2.8e5 < b < 1.0800000000000001e-8

   1. Initial program 96.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. *-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 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -280000 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 75.5

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

4. Applied rewrites 75.5%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 59.4

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

7. Applied rewrites 59.4%

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

9. Applied rewrites 59.4%

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

Alternative 12: 34.9% 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 75.5%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 34.1%

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

Reproduce

herbie shell --seed 2024307 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))