rsin A (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 14.7s

Alternatives: 7

Speedup: 1.0×

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: 76.5% 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

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

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 75.7%

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

   1. associate-/l* 75.8%

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

   2. +-commutative 75.8%

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

3. Simplified 75.8%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 76.5% 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 75.7%

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

   1. +-commutative 75.7%

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

3. Simplified 75.7%

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

   1. *-commutative 75.7%

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

   2. associate-/l* 75.8%

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

6. Applied egg-rr 75.8%

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

Alternative 3: 76.5% 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 75.7%

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

   1. associate-/l* 75.8%

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

   2. +-commutative 75.8%

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

3. Simplified 75.8%

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

Alternative 4: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0082 \lor \neg \left(b \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0082) (not (<= b 3.2e-5)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0082) || !(b <= 3.2e-5)) {
		tmp = r * tan(b);
	} 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 <= (-0.0082d0)) .or. (.not. (b <= 3.2d-5))) then
        tmp = r * tan(b)
    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 <= -0.0082) || !(b <= 3.2e-5)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0082) or not (b <= 3.2e-5):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0082) || !(b <= 3.2e-5))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0082) || ~((b <= 3.2e-5)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0082], N[Not[LessEqual[b, 3.2e-5]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00820000000000000069 or 3.19999999999999986e-5 < b

   1. Initial program 56.9%

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

      1. associate-/l* 56.9%

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

      2. +-commutative 56.9%

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

   3. Simplified 56.9%

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

      1. associate-*r/ 56.9%

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

      2. add-exp-log 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-/l* 26.0%

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

   6. Applied egg-rr 26.0%

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

   7. Taylor expanded in a around 0 25.9%

      \[\leadsto e^{\color{blue}{\log \left(\frac{r \cdot \sin b}{\cos b}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 25.9%

         \[\leadsto e^{\log \left(\frac{\color{blue}{\sin b \cdot r}}{\cos b}\right)} \]

   9. Simplified 25.9%

      \[\leadsto e^{\color{blue}{\log \left(\frac{\sin b \cdot r}{\cos b}\right)}} \]
   10. Step-by-step derivation

       1. rem-exp-log 57.2%

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

       2. *-commutative 57.2%

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

       3. associate-/l* 57.3%

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

       4. add-cube-cbrt 56.6%

          \[\leadsto \color{blue}{\left(\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right) \cdot \sqrt[3]{r}\right)} \cdot \frac{\sin b}{\cos b} \]

       5. unpow2 56.6%

          \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \sqrt[3]{r}\right) \cdot \frac{\sin b}{\cos b} \]

       6. associate-*r* 56.6%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos b}\right)} \]

       7. *-commutative 56.6%

          \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \color{blue}{\left(\frac{\sin b}{\cos b} \cdot \sqrt[3]{r}\right)} \]

       8. *-commutative 56.6%

          \[\leadsto \color{blue}{\left(\frac{\sin b}{\cos b} \cdot \sqrt[3]{r}\right) \cdot {\left(\sqrt[3]{r}\right)}^{2}} \]

       9. associate-*l* 56.6%

          \[\leadsto \color{blue}{\frac{\sin b}{\cos b} \cdot \left(\sqrt[3]{r} \cdot {\left(\sqrt[3]{r}\right)}^{2}\right)} \]

       10. quot-tan 56.5%

           \[\leadsto \color{blue}{\tan b} \cdot \left(\sqrt[3]{r} \cdot {\left(\sqrt[3]{r}\right)}^{2}\right) \]

       11. unpow2 56.5%

           \[\leadsto \tan b \cdot \left(\sqrt[3]{r} \cdot \color{blue}{\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right)}\right) \]

       12. associate-*l* 56.5%

           \[\leadsto \tan b \cdot \color{blue}{\left(\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right) \cdot \sqrt[3]{r}\right)} \]

       13. add-cube-cbrt 57.4%

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

   11. Applied egg-rr 57.4%

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

   if -0.00820000000000000069 < b < 3.19999999999999986e-5

   1. Initial program 98.8%

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

      1. associate-/l* 98.9%

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

      2. +-commutative 98.9%

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

   3. Simplified 98.9%

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

      1. cos-sum 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0 98.6%

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

      1. associate-/l* 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0082 \lor \neg \left(b \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 2.0× 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 75.7%

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

   1. associate-/l* 75.8%

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

   2. +-commutative 75.8%

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

3. Simplified 75.8%

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

   1. associate-*r/ 75.7%

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

   2. add-exp-log 43.2%

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

   3. *-commutative 43.2%

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

   4. associate-/l* 43.2%

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

6. Applied egg-rr 43.2%

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

7. Taylor expanded in a around 0 37.9%

   \[\leadsto e^{\color{blue}{\log \left(\frac{r \cdot \sin b}{\cos b}\right)}} \]
8. Step-by-step derivation

   1. *-commutative 37.9%

      \[\leadsto e^{\log \left(\frac{\color{blue}{\sin b \cdot r}}{\cos b}\right)} \]

9. Simplified 37.9%

   \[\leadsto e^{\color{blue}{\log \left(\frac{\sin b \cdot r}{\cos b}\right)}} \]
10. Step-by-step derivation

    1. rem-exp-log 61.6%

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

    2. *-commutative 61.6%

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

    3. associate-/l* 61.6%

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

    4. add-cube-cbrt 60.9%

       \[\leadsto \color{blue}{\left(\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right) \cdot \sqrt[3]{r}\right)} \cdot \frac{\sin b}{\cos b} \]

    5. unpow2 60.9%

       \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \sqrt[3]{r}\right) \cdot \frac{\sin b}{\cos b} \]

    6. associate-*r* 61.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos b}\right)} \]

    7. *-commutative 61.0%

       \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \color{blue}{\left(\frac{\sin b}{\cos b} \cdot \sqrt[3]{r}\right)} \]

    8. *-commutative 61.0%

       \[\leadsto \color{blue}{\left(\frac{\sin b}{\cos b} \cdot \sqrt[3]{r}\right) \cdot {\left(\sqrt[3]{r}\right)}^{2}} \]

    9. associate-*l* 60.9%

       \[\leadsto \color{blue}{\frac{\sin b}{\cos b} \cdot \left(\sqrt[3]{r} \cdot {\left(\sqrt[3]{r}\right)}^{2}\right)} \]

    10. quot-tan 60.9%

        \[\leadsto \color{blue}{\tan b} \cdot \left(\sqrt[3]{r} \cdot {\left(\sqrt[3]{r}\right)}^{2}\right) \]

    11. unpow2 60.9%

        \[\leadsto \tan b \cdot \left(\sqrt[3]{r} \cdot \color{blue}{\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right)}\right) \]

    12. associate-*l* 60.9%

        \[\leadsto \tan b \cdot \color{blue}{\left(\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right) \cdot \sqrt[3]{r}\right)} \]

    13. add-cube-cbrt 61.6%

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

11. Applied egg-rr 61.6%

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

12. Final simplification 61.6%

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

Alternative 6: 38.8% accurate, 2.0× 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 75.7%

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

   1. +-commutative 75.7%

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

3. Simplified 75.7%

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

   1. associate-*r/ 75.8%

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

   2. add-cube-cbrt 74.9%

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

   3. associate-*l* 74.9%

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

   4. pow2 74.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \]

6. Applied egg-rr 74.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)} \]

7. Taylor expanded in b around 0 50.1%

   \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\color{blue}{\cos a}}\right) \]

8. Taylor expanded in a around 0 36.8%

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

   1. *-commutative 36.8%

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

10. Simplified 36.8%

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

11. Final simplification 36.8%

    \[\leadsto r \cdot \sin b \]
12. Add Preprocessing

Alternative 7: 34.7% accurate, 69.0× 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 75.7%

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

   1. associate-/l* 75.8%

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

   2. +-commutative 75.8%

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

3. Simplified 75.8%

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

5. Taylor expanded in b around 0 46.7%

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

6. Taylor expanded in a around 0 32.3%

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

Reproduce

herbie shell --seed 2024090 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))