rsin B (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 14.4s

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: 77.1% 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, 0.3× speedup

Math

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

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(sin(b) * Float64(-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 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

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 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 78.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

   1. sin-mult 74.2%

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

   2. div-sub 74.2%

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

   3. cos-diff 74.3%

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

   4. add-sqr-sqrt 42.5%

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

   5. sqrt-unprod 73.9%

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

   6. sqr-neg 73.9%

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

   7. sqrt-unprod 31.4%

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

   8. add-sqr-sqrt 72.7%

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

   9. cancel-sign-sub-inv 72.7%

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

   10. cos-sum 74.2%

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

7. Applied egg-rr 74.2%

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

   1. +-inverses 74.2%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]

9. Simplified 74.2%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]

10. Final simplification 74.2%

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

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \lor \neg \left(b \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.0) (not (<= b 5e-16)))
   (* r (/ (sin b) (cos b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.0) || !(b <= 5e-16)) {
		tmp = r * (sin(b) / cos(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 <= (-1.0d0)) .or. (.not. (b <= 5d-16))) then
        tmp = r * (sin(b) / cos(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 <= -1.0) || !(b <= 5e-16)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.0) or not (b <= 5e-16):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.0) || !(b <= 5e-16))
		tmp = Float64(r * Float64(sin(b) / cos(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 <= -1.0) || ~((b <= 5e-16)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.0], N[Not[LessEqual[b, 5e-16]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1 or 5.0000000000000004e-16 < b

   1. Initial program 51.2%

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

      1. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in a around 0 51.6%

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

   if -1 < b < 5.0000000000000004e-16

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 98.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \lor \neg \left(b \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 5: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.0)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 5e-16) (* r (/ b (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 5e-16) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r * (sin(b) / cos(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 <= (-1.0d0)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 5d-16) then
        tmp = r * (b / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 5e-16) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.0:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 5e-16:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.0)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 5e-16)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.0)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 5e-16)
		tmp = r * (b / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.0], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-16], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1

   1. Initial program 53.4%

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

      1. *-commutative 53.4%

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

      2. associate-/r/ 53.3%

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

      3. +-commutative 53.3%

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

   3. Simplified 53.3%

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

      1. add-cube-cbrt 52.8%

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

      2. pow3 52.7%

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

   5. Applied egg-rr 52.7%

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

   6. Taylor expanded in a around 0 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-/l* 54.6%

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

   8. Simplified 54.6%

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

   if -1 < b < 5.0000000000000004e-16

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 98.7%

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

   if 5.0000000000000004e-16 < b

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in a around 0 49.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;r \cdot \frac{1}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.0)
   (* r (/ 1.0 (/ (cos b) (sin b))))
   (if (<= b 5e-16) (* r (/ b (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = r * (1.0 / (cos(b) / sin(b)));
	} else if (b <= 5e-16) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r * (sin(b) / cos(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 <= (-1.0d0)) then
        tmp = r * (1.0d0 / (cos(b) / sin(b)))
    else if (b <= 5d-16) then
        tmp = r * (b / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = r * (1.0 / (Math.cos(b) / Math.sin(b)));
	} else if (b <= 5e-16) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.0:
		tmp = r * (1.0 / (math.cos(b) / math.sin(b)))
	elif b <= 5e-16:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.0)
		tmp = Float64(r * Float64(1.0 / Float64(cos(b) / sin(b))));
	elseif (b <= 5e-16)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.0)
		tmp = r * (1.0 / (cos(b) / sin(b)));
	elseif (b <= 5e-16)
		tmp = r * (b / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.0], N[(r * N[(1.0 / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-16], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1

   1. Initial program 53.4%

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

      1. +-commutative 53.4%

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

   3. Simplified 53.4%

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

      1. clear-num 53.4%

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

      2. inv-pow 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. unpow-1 53.4%

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

      2. +-commutative 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in a around 0 54.6%

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

   if -1 < b < 5.0000000000000004e-16

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 98.7%

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

   if 5.0000000000000004e-16 < b

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in a around 0 49.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;r \cdot \frac{1}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 7: 77.1% 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 72.9%

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

2. Final simplification 72.9%

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

Alternative 8: 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 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. add-sqr-sqrt 41.6%

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

   3. sqrt-unprod 83.3%

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

   4. sqr-neg 83.3%

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

   5. sqrt-unprod 41.8%

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

   6. add-sqr-sqrt 72.8%

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

   7. cos-diff 73.0%

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

7. Applied egg-rr 73.0%

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

8. Final simplification 73.0%

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

Alternative 9: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{+31} \lor \neg \left(b \leq 10200000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -9.6e+31) (not (<= b 10200000000000.0)))
   (* r (sin b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9.6e+31) || !(b <= 10200000000000.0)) {
		tmp = r * sin(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 <= (-9.6d+31)) .or. (.not. (b <= 10200000000000.0d0))) then
        tmp = r * sin(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 <= -9.6e+31) || !(b <= 10200000000000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -9.6e+31) or not (b <= 10200000000000.0):
		tmp = r * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -9.6e+31) || !(b <= 10200000000000.0))
		tmp = Float64(r * sin(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 <= -9.6e+31) || ~((b <= 10200000000000.0)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -9.6e+31], N[Not[LessEqual[b, 10200000000000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.59999999999999929e31 or 1.02e13 < b

   1. Initial program 49.8%

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

      1. +-commutative 49.8%

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

   3. Simplified 49.8%

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

      1. cos-sum 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. fma-def 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in a around 0 52.2%

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

   7. Taylor expanded in b around 0 12.1%

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

   if -9.59999999999999929e31 < b < 1.02e13

   1. Initial program 96.4%

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

      1. +-commutative 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in b around 0 92.7%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{+31} \lor \neg \left(b \leq 10200000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

4. Taylor expanded in b around 0 47.6%

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

5. Final simplification 47.6%

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

Alternative 11: 34.5% 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 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

4. Taylor expanded in b around 0 47.6%

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

5. Taylor expanded in a around 0 30.9%

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

6. Final simplification 30.9%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023200 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))