rsin A (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 11.8s

Alternatives: 7

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% 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 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 77.0% 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 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

Alternative 3: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00068 \lor \neg \left(b \leq 0.00056\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00068) (not (<= b 0.00056)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00068) || !(b <= 0.00056)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / 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 <= (-0.00068d0)) .or. (.not. (b <= 0.00056d0))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00068) || !(b <= 0.00056)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00068) or not (b <= 0.00056):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00068) || !(b <= 0.00056))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00068) || ~((b <= 0.00056)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00068], N[Not[LessEqual[b, 0.00056]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.8e-4 or 5.5999999999999995e-4 < b

   1. Initial program 55.8%

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

      1. associate-/l* 55.8%

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

      2. remove-double-neg 55.8%

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

      3. remove-double-neg 55.8%

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

      4. +-commutative 55.8%

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

   3. Simplified 55.8%

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

      1. add-sqr-sqrt 33.2%

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

      2. sqrt-unprod 37.1%

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

      3. pow2 37.1%

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

   6. Applied egg-rr 37.1%

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

      1. sqrt-pow1 55.8%

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

      2. metadata-eval 55.8%

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

      3. pow1 55.8%

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

      4. clear-num 55.7%

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

   8. Applied egg-rr 55.7%

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

   9. Taylor expanded in a around 0 56.3%

      \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\cos b}{\sin b}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 56.3%

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

       2. clear-num 56.3%

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

       3. quot-tan 56.4%

          \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   11. Applied egg-rr 56.4%

       \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]

   13. Simplified 56.4%

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

   if -6.8e-4 < b < 5.5999999999999995e-4

   1. Initial program 98.5%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. remove-double-neg 98.6%

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

      4. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in b around 0 98.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00068 \lor \neg \left(b \leq 0.00056\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00064 \lor \neg \left(b \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00064) (not (<= b 4.8e-6)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00064) || !(b <= 4.8e-6)) {
		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 <= (-0.00064d0)) .or. (.not. (b <= 4.8d-6))) 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 <= -0.00064) || !(b <= 4.8e-6)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00064) or not (b <= 4.8e-6):
		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 <= -0.00064) || !(b <= 4.8e-6))
		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 <= -0.00064) || ~((b <= 4.8e-6)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00064], N[Not[LessEqual[b, 4.8e-6]], $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 < -6.40000000000000052e-4 or 4.7999999999999998e-6 < b

   1. Initial program 55.8%

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

      1. associate-/l* 55.8%

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

      2. remove-double-neg 55.8%

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

      3. remove-double-neg 55.8%

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

      4. +-commutative 55.8%

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

   3. Simplified 55.8%

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

      1. add-sqr-sqrt 33.2%

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

      2. sqrt-unprod 37.1%

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

      3. pow2 37.1%

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

   6. Applied egg-rr 37.1%

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

      1. sqrt-pow1 55.8%

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

      2. metadata-eval 55.8%

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

      3. pow1 55.8%

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

      4. clear-num 55.7%

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

   8. Applied egg-rr 55.7%

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

   9. Taylor expanded in a around 0 56.3%

      \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\cos b}{\sin b}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 56.3%

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

       2. clear-num 56.3%

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

       3. quot-tan 56.4%

          \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   11. Applied egg-rr 56.4%

       \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]

   13. Simplified 56.4%

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

   if -6.40000000000000052e-4 < b < 4.7999999999999998e-6

   1. Initial program 98.5%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. remove-double-neg 98.6%

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

      4. +-commutative 98.6%

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

   3. Simplified 98.6%

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

      1. cos-sum 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around 0 99.8%

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

      1. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in b around 0 98.5%

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

       1. associate-/l* 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00064 \lor \neg \left(b \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00064:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.00064)
   (* r (tan b))
   (if (<= b 1.85e-5) (* r (/ b (cos a))) (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.00064) {
		tmp = r * tan(b);
	} else if (b <= 1.85e-5) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / (1.0 / 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 <= (-0.00064d0)) then
        tmp = r * tan(b)
    else if (b <= 1.85d-5) then
        tmp = r * (b / cos(a))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.00064) {
		tmp = r * Math.tan(b);
	} else if (b <= 1.85e-5) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.00064:
		tmp = r * math.tan(b)
	elif b <= 1.85e-5:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.00064)
		tmp = Float64(r * tan(b));
	elseif (b <= 1.85e-5)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.00064)
		tmp = r * tan(b);
	elseif (b <= 1.85e-5)
		tmp = r * (b / cos(a));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.00064], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e-5], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.40000000000000052e-4

   1. Initial program 52.1%

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

      1. associate-/l* 52.0%

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

      2. remove-double-neg 52.0%

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

      3. remove-double-neg 52.0%

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

      4. +-commutative 52.0%

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

   3. Simplified 52.0%

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

      1. add-sqr-sqrt 31.4%

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

      2. sqrt-unprod 36.0%

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

      3. pow2 36.0%

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

   6. Applied egg-rr 36.0%

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

      1. sqrt-pow1 52.0%

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

      2. metadata-eval 52.0%

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

      3. pow1 52.0%

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

      4. clear-num 51.9%

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

   8. Applied egg-rr 51.9%

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

   9. Taylor expanded in a around 0 52.6%

      \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\cos b}{\sin b}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 52.6%

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

       2. clear-num 52.7%

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

       3. quot-tan 52.8%

          \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   11. Applied egg-rr 52.8%

       \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]

   13. Simplified 52.8%

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

   if -6.40000000000000052e-4 < b < 1.84999999999999991e-5

   1. Initial program 98.5%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. remove-double-neg 98.6%

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

      4. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in b around 0 98.6%

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

   if 1.84999999999999991e-5 < b

   1. Initial program 59.3%

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

      1. associate-/l* 59.2%

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

      2. remove-double-neg 59.2%

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

      3. remove-double-neg 59.2%

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

      4. +-commutative 59.2%

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

   3. Simplified 59.2%

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

      1. add-sqr-sqrt 34.8%

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

      2. sqrt-unprod 38.2%

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

      3. pow2 38.2%

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

   6. Applied egg-rr 38.2%

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

      1. sqrt-pow1 59.2%

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

      2. metadata-eval 59.2%

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

      3. pow1 59.2%

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

      4. clear-num 59.2%

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

   8. Applied egg-rr 59.2%

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

   9. Taylor expanded in a around 0 59.7%

      \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\cos b}{\sin b}}} \]
   10. Step-by-step derivation

       1. un-div-inv 59.8%

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

       2. clear-num 59.7%

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

       3. quot-tan 59.8%

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

   11. Applied egg-rr 59.8%

       \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.5% 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 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. add-sqr-sqrt 44.0%

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

   2. sqrt-unprod 43.8%

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

   3. pow2 43.8%

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

6. Applied egg-rr 43.8%

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

   1. sqrt-pow1 77.4%

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

   2. metadata-eval 77.4%

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

   3. pow1 77.4%

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

   4. clear-num 77.2%

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

8. Applied egg-rr 77.2%

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

9. Taylor expanded in a around 0 60.0%

   \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\cos b}{\sin b}}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 60.0%

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

    2. clear-num 60.1%

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

    3. quot-tan 60.2%

       \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

11. Applied egg-rr 60.2%

    \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]

13. Simplified 60.2%

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

Alternative 7: 34.9% 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 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

5. Taylor expanded in b around 0 51.6%

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

6. Taylor expanded in a around 0 34.0%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))