rsin B (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 13.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. cos-sum N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 78.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 78.4

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

4. Applied egg-rr 78.4%

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

5. Final simplification 78.4%

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

Alternative 4: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Final simplification 78.3%

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

Alternative 5: 76.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0032:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 0.0017:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0032)
   (/ r (/ 1.0 (tan b)))
   (if (<= b 0.0017)
     (* r (/ (fma b (* b (* b -0.16666666666666666)) b) (cos (+ b a))))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0032) {
		tmp = r / (1.0 / tan(b));
	} else if (b <= 0.0017) {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / cos((b + a)));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0032)
		tmp = Float64(r / Float64(1.0 / tan(b)));
	elseif (b <= 0.0017)
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / cos(Float64(b + a))));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0032], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0017], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00320000000000000015

   1. Initial program 51.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 52.3

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

   5. Simplified 52.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 52.3

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. quot-tan N/A

          \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

      13. lower-tan.f64 52.4

          \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   7. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   if -0.00320000000000000015 < b < 0.00169999999999999991

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.1

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

   5. Simplified 98.1%

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

   if 0.00169999999999999991 < b

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 66.1

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

   5. Simplified 66.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.1

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

      6. lift-/.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. quot-tan N/A

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

      10. lower-tan.f64 66.5

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

   7. Applied egg-rr 66.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0032:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 0.0017:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -6e-5)
   (/ r (/ 1.0 (tan b)))
   (if (<= b 7.2e-5) (* b (/ r (cos a))) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -6e-5) {
		tmp = r / (1.0 / tan(b));
	} else if (b <= 7.2e-5) {
		tmp = b * (r / cos(a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6d-5)) then
        tmp = r / (1.0d0 / tan(b))
    else if (b <= 7.2d-5) then
        tmp = b * (r / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -6e-5) {
		tmp = r / (1.0 / Math.tan(b));
	} else if (b <= 7.2e-5) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -6e-5:
		tmp = r / (1.0 / math.tan(b))
	elif b <= 7.2e-5:
		tmp = b * (r / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -6e-5)
		tmp = Float64(r / Float64(1.0 / tan(b)));
	elseif (b <= 7.2e-5)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -6e-5)
		tmp = r / (1.0 / tan(b));
	elseif (b <= 7.2e-5)
		tmp = b * (r / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -6e-5], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-5], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000015e-5

   1. Initial program 51.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 52.3

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

   5. Simplified 52.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 52.3

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. quot-tan N/A

          \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

      13. lower-tan.f64 52.4

          \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   7. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   if -6.00000000000000015e-5 < b < 7.20000000000000018e-5

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 97.9

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

   5. Simplified 97.9%

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

   if 7.20000000000000018e-5 < b

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 66.1

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

   5. Simplified 66.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.1

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

      6. lift-/.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. quot-tan N/A

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

      10. lower-tan.f64 66.5

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

   7. Applied egg-rr 66.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -6 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -6e-5) t_0 (if (<= b 7.2e-5) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -6e-5) {
		tmp = t_0;
	} else if (b <= 7.2e-5) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-6d-5)) then
        tmp = t_0
    else if (b <= 7.2d-5) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -6e-5) {
		tmp = t_0;
	} else if (b <= 7.2e-5) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -6e-5:
		tmp = t_0
	elif b <= 7.2e-5:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -6e-5)
		tmp = t_0;
	elseif (b <= 7.2e-5)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -6e-5)
		tmp = t_0;
	elseif (b <= 7.2e-5)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6e-5], t$95$0, If[LessEqual[b, 7.2e-5], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.00000000000000015e-5 or 7.20000000000000018e-5 < b

   1. Initial program 59.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 60.2

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

   5. Simplified 60.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.2

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

      6. lift-/.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. quot-tan N/A

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

      10. lower-tan.f64 60.3

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

   7. Applied egg-rr 60.3%

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

   if -6.00000000000000015e-5 < b < 7.20000000000000018e-5

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 97.9

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

   5. Simplified 97.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 63.4

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

5. Simplified 63.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 63.4

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

   6. lift-/.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. quot-tan N/A

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

   10. lower-tan.f64 63.4

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

7. Applied egg-rr 63.4%

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

8. Final simplification 63.4%

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

Alternative 9: 38.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. flip3-+ N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{a \cdot a + \left(b \cdot b - a \cdot b\right)}\right)}} \]

   2. frac-2neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left({a}^{3} + {b}^{3}\right)\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

   3. distribute-frac-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)\right)}} \]

   4. cos-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

   5. lower-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

   7. cube-mult N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \left(\frac{\color{blue}{a \cdot \left(a \cdot a\right)} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \left(\frac{\color{blue}{\mathsf{fma}\left(a, a \cdot a, {b}^{3}\right)}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \left(\frac{\mathsf{fma}\left(a, \color{blue}{a \cdot a}, {b}^{3}\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

   10. cube-mult N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. associate-+r- N/A

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

   15. +-commutative N/A

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

   16. associate-+r- N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

4. Applied egg-rr 38.8%

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

5. Taylor expanded in a around inf

   \[\leadsto r \cdot \frac{\sin b}{\cos \left(\frac{\color{blue}{{a}^{3}}}{\mathsf{neg}\left(\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)\right)}\right)} \]
6. Step-by-step derivation

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 33.1

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

7. Simplified 33.1%

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

8. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 38.3

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

10. Simplified 38.3%

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

Alternative 10: 35.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{1}{\frac{\mathsf{fma}\left(b, b \cdot -0.3333333333333333, 1\right)}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ 1.0 (/ (fma b (* b -0.3333333333333333) 1.0) b))))

C

double code(double r, double a, double b) {
	return r * (1.0 / (fma(b, (b * -0.3333333333333333), 1.0) / b));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(1.0 / Float64(fma(b, Float64(b * -0.3333333333333333), 1.0) / b)))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(1.0 / N[(N[(b * N[(b * -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 63.4

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

5. Simplified 63.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. lower-/.f64 63.3

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

   8. lift-/.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. quot-tan N/A

       \[\leadsto r \cdot \frac{1}{\frac{1}{\color{blue}{\tan b}}} \]

   12. lower-tan.f64 63.4

       \[\leadsto r \cdot \frac{1}{\frac{1}{\color{blue}{\tan b}}} \]

7. Applied egg-rr 63.4%

   \[\leadsto r \cdot \color{blue}{\frac{1}{\frac{1}{\tan b}}} \]

8. Taylor expanded in b around 0

   \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{1 + \frac{-1}{3} \cdot {b}^{2}}{b}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{1 + \frac{-1}{3} \cdot {b}^{2}}{b}}} \]

   2. +-commutative N/A

      \[\leadsto r \cdot \frac{1}{\frac{\color{blue}{\frac{-1}{3} \cdot {b}^{2} + 1}}{b}} \]

   3. *-commutative N/A

      \[\leadsto r \cdot \frac{1}{\frac{\color{blue}{{b}^{2} \cdot \frac{-1}{3}} + 1}{b}} \]

   4. unpow2 N/A

      \[\leadsto r \cdot \frac{1}{\frac{\color{blue}{\left(b \cdot b\right)} \cdot \frac{-1}{3} + 1}{b}} \]

   5. associate-*l* N/A

      \[\leadsto r \cdot \frac{1}{\frac{\color{blue}{b \cdot \left(b \cdot \frac{-1}{3}\right)} + 1}{b}} \]

   6. lower-fma.f64 N/A

      \[\leadsto r \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(b, b \cdot \frac{-1}{3}, 1\right)}}{b}} \]

   7. lower-*.f64 34.9

      \[\leadsto r \cdot \frac{1}{\frac{\mathsf{fma}\left(b, \color{blue}{b \cdot -0.3333333333333333}, 1\right)}{b}} \]

10. Simplified 34.9%

    \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(b, b \cdot -0.3333333333333333, 1\right)}{b}}} \]
11. Add Preprocessing

Alternative 11: 34.7% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 49.4

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

5. Simplified 49.4%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 34.1

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

8. Simplified 34.1%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))