rsin B (should all be same)

Percentage Accurate: 76.5% → 99.4%

Time: 18.5s

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: 76.5% 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return sin(b) * (r / ((cos(a) * cos(b)) - (sin(a) * 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 = sin(b) * (r / ((cos(a) * cos(b)) - (sin(a) * sin(b))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. +-commutative 78.0%

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

3. Simplified 78.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. prod-diff 99.5%

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

7. Applied egg-rr 99.5%

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

   1. fma-udef 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. fma-udef 99.5%

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

   8. *-rgt-identity 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. *-rgt-identity 99.5%

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

   11. fma-udef 99.5%

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

   12. *-commutative 99.5%

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

9. Simplified 99.5%

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

    1. *-commutative 99.5%

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

    2. add-sqr-sqrt 45.9%

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

    3. sqrt-unprod 89.0%

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

    4. sqr-neg 89.0%

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

    5. sqrt-unprod 43.1%

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

    6. add-sqr-sqrt 77.7%

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

    7. log1p-expm1-u 77.7%

       \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin b\right) \cdot \sin a\right)\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

    8. add-sqr-sqrt 43.1%

       \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{-\sin b} \cdot \sqrt{-\sin b}\right)} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

    9. sqrt-unprod 89.0%

       \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-\sin b\right) \cdot \left(-\sin b\right)}} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

    10. sqr-neg 89.0%

        \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{\sin b \cdot \sin b}} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

    11. sqrt-unprod 45.8%

        \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{\sin b} \cdot \sqrt{\sin b}\right)} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

    12. add-sqr-sqrt 99.5%

        \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sin b} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

11. Applied egg-rr 99.5%

    \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

12. Taylor expanded in r around 0 99.5%

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

    1. associate-/l* 99.4%

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

    2. associate-/r/ 99.5%

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

    3. +-commutative 99.5%

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

    4. mul-1-neg 99.5%

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

    5. unsub-neg 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. +-commutative 78.0%

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

3. Simplified 78.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00052 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00052) (not (<= a 1.15e-14)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00052) || !(a <= 1.15e-14)) {
		tmp = r * (sin(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 ((a <= (-0.00052d0)) .or. (.not. (a <= 1.15d-14))) then
        tmp = r * (sin(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 ((a <= -0.00052) || !(a <= 1.15e-14)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00052) or not (a <= 1.15e-14):
		tmp = r * (math.sin(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 ((a <= -0.00052) || !(a <= 1.15e-14))
		tmp = Float64(r * Float64(sin(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 ((a <= -0.00052) || ~((a <= 1.15e-14)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00052], N[Not[LessEqual[a, 1.15e-14]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.19999999999999954e-4 or 1.14999999999999999e-14 < a

   1. Initial program 58.7%

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

      1. +-commutative 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in b around 0 58.7%

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

   if -5.19999999999999954e-4 < a < 1.14999999999999999e-14

   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 a around 0 98.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00052 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00066 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00066) (not (<= a 1.15e-14)))
   (* r (/ (sin b) (cos a)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00066) || !(a <= 1.15e-14)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = sin(b) * (r / 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 ((a <= (-0.00066d0)) .or. (.not. (a <= 1.15d-14))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00066) || !(a <= 1.15e-14)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00066) or not (a <= 1.15e-14):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00066) || !(a <= 1.15e-14))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00066) || ~((a <= 1.15e-14)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00066], N[Not[LessEqual[a, 1.15e-14]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6e-4 or 1.14999999999999999e-14 < a

   1. Initial program 58.7%

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

      1. +-commutative 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in b around 0 58.7%

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

   if -6.6e-4 < a < 1.14999999999999999e-14

   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. Step-by-step derivation

      1. cos-sum 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. prod-diff 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-lft-neg-in 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-udef 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. fma-udef 99.7%

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

      12. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. *-commutative 99.7%

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

       2. add-sqr-sqrt 46.2%

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

       3. sqrt-unprod 99.5%

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

       4. sqr-neg 99.5%

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

       5. sqrt-unprod 53.3%

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

       6. add-sqr-sqrt 98.7%

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

       7. log1p-expm1-u 98.7%

          \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin b\right) \cdot \sin a\right)\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       8. add-sqr-sqrt 53.3%

          \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{-\sin b} \cdot \sqrt{-\sin b}\right)} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       9. sqrt-unprod 99.5%

          \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-\sin b\right) \cdot \left(-\sin b\right)}} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       10. sqr-neg 99.5%

           \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{\sin b \cdot \sin b}} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       11. sqrt-unprod 46.2%

           \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{\sin b} \cdot \sqrt{\sin b}\right)} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       12. add-sqr-sqrt 99.7%

           \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sin b} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

   11. Applied egg-rr 99.7%

       \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

   12. Taylor expanded in a around 0 98.6%

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

       1. associate-/l* 98.5%

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

       2. associate-/r/ 98.7%

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

   14. Simplified 98.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00066 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 5: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00136 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00136) (not (<= a 1.15e-14)))
   (/ (sin b) (/ (cos a) r))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00136) || !(a <= 1.15e-14)) {
		tmp = sin(b) / (cos(a) / r);
	} else {
		tmp = sin(b) * (r / 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 ((a <= (-0.00136d0)) .or. (.not. (a <= 1.15d-14))) then
        tmp = sin(b) / (cos(a) / r)
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00136) || !(a <= 1.15e-14)) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00136) or not (a <= 1.15e-14):
		tmp = math.sin(b) / (math.cos(a) / r)
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00136) || !(a <= 1.15e-14))
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00136) || ~((a <= 1.15e-14)))
		tmp = sin(b) / (cos(a) / r);
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00136], N[Not[LessEqual[a, 1.15e-14]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00136 or 1.14999999999999999e-14 < a

   1. Initial program 58.7%

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

      1. +-commutative 58.7%

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

   3. Simplified 58.7%

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

      1. associate-*r/ 58.7%

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

      2. clear-num 58.2%

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

      3. *-commutative 58.2%

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

   5. Applied egg-rr 58.2%

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

   6. Taylor expanded in b around inf 58.7%

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

      1. *-commutative 58.7%

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

      2. +-commutative 58.7%

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

      3. associate-/l* 58.8%

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

   8. Simplified 58.8%

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

   9. Taylor expanded in b around 0 58.7%

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

   if -0.00136 < a < 1.14999999999999999e-14

   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. Step-by-step derivation

      1. cos-sum 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. prod-diff 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-lft-neg-in 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-udef 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. fma-udef 99.7%

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

      12. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. *-commutative 99.7%

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

       2. add-sqr-sqrt 46.2%

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

       3. sqrt-unprod 99.5%

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

       4. sqr-neg 99.5%

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

       5. sqrt-unprod 53.3%

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

       6. add-sqr-sqrt 98.7%

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

       7. log1p-expm1-u 98.7%

          \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin b\right) \cdot \sin a\right)\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       8. add-sqr-sqrt 53.3%

          \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{-\sin b} \cdot \sqrt{-\sin b}\right)} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       9. sqrt-unprod 99.5%

          \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-\sin b\right) \cdot \left(-\sin b\right)}} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       10. sqr-neg 99.5%

           \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{\sin b \cdot \sin b}} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       11. sqrt-unprod 46.2%

           \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{\sin b} \cdot \sqrt{\sin b}\right)} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

       12. add-sqr-sqrt 99.7%

           \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sin b} \cdot \sin a\right)\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

   11. Applied egg-rr 99.7%

       \[\leadsto r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

   12. Taylor expanded in a around 0 98.6%

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

       1. associate-/l* 98.5%

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

       2. associate-/r/ 98.7%

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

   14. Simplified 98.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00136 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6: 76.5% 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]

Derivation

1. Initial program 78.0%

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

2. Final simplification 78.0%

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

Alternative 7: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. +-commutative 78.0%

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

3. Simplified 78.0%

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

4. Taylor expanded in b around 0 55.7%

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

5. Final simplification 55.7%

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

Alternative 8: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 9200000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.68) (not (<= b 9200000.0)))
   (* r (sin b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.68) || !(b <= 9200000.0)) {
		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 <= (-0.68d0)) .or. (.not. (b <= 9200000.0d0))) 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 <= -0.68) || !(b <= 9200000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.68) or not (b <= 9200000.0):
		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 <= -0.68) || !(b <= 9200000.0))
		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 <= -0.68) || ~((b <= 9200000.0)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.68], N[Not[LessEqual[b, 9200000.0]], $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 < -0.680000000000000049 or 9.2e6 < b

   1. Initial program 55.8%

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

      1. +-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. Step-by-step derivation

      1. add-sqr-sqrt 31.7%

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

      2. sqrt-unprod 34.8%

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

      3. pow2 34.8%

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

   5. Applied egg-rr 34.8%

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

   6. Taylor expanded in b around 0 11.6%

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

   7. Taylor expanded in a around 0 11.1%

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

   if -0.680000000000000049 < b < 9.2e6

   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 97.6%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 9200000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 9: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 8200000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.68) (not (<= b 8200000.0)))
   (* r (sin b))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.68) || !(b <= 8200000.0)) {
		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 <= (-0.68d0)) .or. (.not. (b <= 8200000.0d0))) 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 <= -0.68) || !(b <= 8200000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.68) or not (b <= 8200000.0):
		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 <= -0.68) || !(b <= 8200000.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.68) || ~((b <= 8200000.0)))
		tmp = r * sin(b);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.68], N[Not[LessEqual[b, 8200000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.680000000000000049 or 8.2e6 < b

   1. Initial program 55.8%

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

      1. +-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. Step-by-step derivation

      1. add-sqr-sqrt 31.7%

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

      2. sqrt-unprod 34.8%

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

      3. pow2 34.8%

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

   5. Applied egg-rr 34.8%

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

   6. Taylor expanded in b around 0 11.6%

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

   7. Taylor expanded in a around 0 11.1%

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

   if -0.680000000000000049 < b < 8.2e6

   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 97.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 8200000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 10: 38.4% 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 78.0%

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

   1. +-commutative 78.0%

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

3. Simplified 78.0%

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

   1. add-sqr-sqrt 43.1%

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

   2. sqrt-unprod 39.1%

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

   3. pow2 39.1%

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

5. Applied egg-rr 39.1%

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

6. Taylor expanded in b around 0 27.5%

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

7. Taylor expanded in a around 0 37.4%

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

8. Final simplification 37.4%

   \[\leadsto r \cdot \sin b \]

Alternative 11: 34.3% 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 78.0%

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

   1. +-commutative 78.0%

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

3. Simplified 78.0%

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

4. Taylor expanded in b around 0 52.0%

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

   1. associate-/l* 51.9%

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

   2. associate-/r/ 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in a around 0 33.8%

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

8. Final simplification 33.8%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023279 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))