rsin A (should all be same)

Percentage Accurate: 75.5% → 99.5%

Time: 18.0s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. r = -9.243665512169372e+304
  2. a = 3.0248908496213486e-151
  3. b = 2.5986723089242488e+237

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: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 75.5%

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

   2. remove-double-neg 75.5%

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

   3. remove-double-neg 75.5%

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

   4. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in r around 0 99.6%

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

   1. associate-/l* 99.5%

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

   2. /-rgt-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. *-commutative 99.5%

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

   5. *-rgt-identity 99.5%

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

   6. cancel-sign-sub-inv 99.5%

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

   7. distribute-lft-neg-in 99.5%

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

   8. fma-undefine 99.5%

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

   9. times-frac 99.6%

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

   10. *-commutative 99.6%

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

   11. times-frac 99.6%

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

   12. /-rgt-identity 99.6%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.25e-8)
   (* r (tan b))
   (if (<= b 2.6) (* (sin b) (/ r (cos a))) (/ 1.0 (/ (/ 1.0 r) (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = r * tan(b);
	} else if (b <= 2.6) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = 1.0 / ((1.0 / r) / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.25d-8)) then
        tmp = r * tan(b)
    else if (b <= 2.6d0) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = 1.0d0 / ((1.0d0 / r) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = r * Math.tan(b);
	} else if (b <= 2.6) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = 1.0 / ((1.0 / r) / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.25e-8:
		tmp = r * math.tan(b)
	elif b <= 2.6:
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = 1.0 / ((1.0 / r) / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.25e-8)
		tmp = Float64(r * tan(b));
	elseif (b <= 2.6)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.25e-8)
		tmp = r * tan(b);
	elseif (b <= 2.6)
		tmp = sin(b) * (r / cos(a));
	else
		tmp = 1.0 / ((1.0 / r) / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.25e-8], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.24999999999999996e-8

   1. Initial program 60.2%

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

      1. associate-/l* 60.1%

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

      2. remove-double-neg 60.1%

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

      3. remove-double-neg 60.1%

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

      4. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. associate-*r/ 60.2%

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

      2. clear-num 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

      1. div-inv 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*r/ 60.0%

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

       2. *-rgt-identity 60.0%

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

       3. associate-/r* 60.0%

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

   11. Simplified 60.0%

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

       1. associate-/l/ 60.0%

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

       2. *-commutative 60.0%

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

       3. clear-num 60.2%

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

       4. associate-/l* 60.1%

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

       5. quot-tan 60.2%

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

   13. Applied egg-rr 60.2%

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

   if -2.24999999999999996e-8 < b < 2.60000000000000009

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

      1. cos-sum 99.8%

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

   6. Applied egg-rr 99.8%

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

   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. associate-/l* 99.8%

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

      2. /-rgt-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. cancel-sign-sub-inv 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. fma-undefine 99.8%

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

      9. times-frac 99.8%

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

      10. *-commutative 99.8%

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

      11. times-frac 99.8%

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

      12. /-rgt-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in b around 0 98.4%

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

   if 2.60000000000000009 < b

   1. Initial program 47.9%

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

      1. associate-/l* 47.9%

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

      2. remove-double-neg 47.9%

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

      3. remove-double-neg 47.9%

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

      4. +-commutative 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*r/ 47.9%

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

      2. clear-num 48.0%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in a around 0 48.6%

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

      1. div-inv 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. associate-*r/ 48.6%

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

       2. *-rgt-identity 48.6%

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

       3. associate-/r* 48.5%

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

   11. Simplified 48.5%

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

       1. div-inv 48.5%

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

       2. *-un-lft-identity 48.5%

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

       3. div-inv 48.5%

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

       4. associate-/l/ 48.6%

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

       5. *-commutative 48.6%

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

       6. clear-num 48.5%

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

       7. associate-/l* 48.5%

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

       8. quot-tan 48.5%

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

   13. Applied egg-rr 48.5%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{1}{r \cdot \tan b}}} \]
   14. Step-by-step derivation

       1. *-lft-identity 48.5%

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

       2. associate-/r* 48.5%

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

   15. Simplified 48.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.25e-8)
   (* (sin b) (/ r (cos b)))
   (if (<= b 2.6) (* (sin b) (/ r (cos a))) (/ 1.0 (/ (/ 1.0 r) (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 2.6) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = 1.0 / ((1.0 / r) / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.25d-8)) then
        tmp = sin(b) * (r / cos(b))
    else if (b <= 2.6d0) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = 1.0d0 / ((1.0d0 / r) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (b <= 2.6) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = 1.0 / ((1.0 / r) / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.25e-8:
		tmp = math.sin(b) * (r / math.cos(b))
	elif b <= 2.6:
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = 1.0 / ((1.0 / r) / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.25e-8)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 2.6)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.25e-8)
		tmp = sin(b) * (r / cos(b));
	elseif (b <= 2.6)
		tmp = sin(b) * (r / cos(a));
	else
		tmp = 1.0 / ((1.0 / r) / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.25e-8], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.24999999999999996e-8

   1. Initial program 60.2%

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

      1. associate-/l* 60.1%

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

      2. remove-double-neg 60.1%

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

      3. remove-double-neg 60.1%

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

      4. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. associate-*r/ 60.2%

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

      2. clear-num 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

      1. div-inv 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*r/ 60.0%

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

       2. *-rgt-identity 60.0%

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

       3. associate-/r* 60.0%

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

   11. Simplified 60.0%

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

       1. associate-/r/ 60.0%

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

       2. clear-num 60.2%

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

   13. Applied egg-rr 60.2%

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

   if -2.24999999999999996e-8 < b < 2.60000000000000009

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

      1. cos-sum 99.8%

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

   6. Applied egg-rr 99.8%

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

   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. associate-/l* 99.8%

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

      2. /-rgt-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. cancel-sign-sub-inv 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. fma-undefine 99.8%

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

      9. times-frac 99.8%

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

      10. *-commutative 99.8%

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

      11. times-frac 99.8%

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

      12. /-rgt-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in b around 0 98.4%

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

   if 2.60000000000000009 < b

   1. Initial program 47.9%

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

      1. associate-/l* 47.9%

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

      2. remove-double-neg 47.9%

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

      3. remove-double-neg 47.9%

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

      4. +-commutative 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*r/ 47.9%

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

      2. clear-num 48.0%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in a around 0 48.6%

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

      1. div-inv 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. associate-*r/ 48.6%

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

       2. *-rgt-identity 48.6%

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

       3. associate-/r* 48.5%

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

   11. Simplified 48.5%

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

       1. div-inv 48.5%

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

       2. *-un-lft-identity 48.5%

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

       3. div-inv 48.5%

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

       4. associate-/l/ 48.6%

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

       5. *-commutative 48.6%

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

       6. clear-num 48.5%

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

       7. associate-/l* 48.5%

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

       8. quot-tan 48.5%

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

   13. Applied egg-rr 48.5%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{1}{r \cdot \tan b}}} \]
   14. Step-by-step derivation

       1. *-lft-identity 48.5%

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

       2. associate-/r* 48.5%

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

   15. Simplified 48.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.25e-8)
   (* (sin b) (/ r (cos b)))
   (if (<= b 2.6) (* (sin b) (/ r (cos a))) (/ (sin b) (/ (cos b) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 2.6) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = sin(b) / (cos(b) / r);
	}
	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 <= (-2.25d-8)) then
        tmp = sin(b) * (r / cos(b))
    else if (b <= 2.6d0) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = sin(b) / (cos(b) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (b <= 2.6) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.25e-8:
		tmp = math.sin(b) * (r / math.cos(b))
	elif b <= 2.6:
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = math.sin(b) / (math.cos(b) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.25e-8)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 2.6)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.25e-8)
		tmp = sin(b) * (r / cos(b));
	elseif (b <= 2.6)
		tmp = sin(b) * (r / cos(a));
	else
		tmp = sin(b) / (cos(b) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.25e-8], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.24999999999999996e-8

   1. Initial program 60.2%

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

      1. associate-/l* 60.1%

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

      2. remove-double-neg 60.1%

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

      3. remove-double-neg 60.1%

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

      4. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. associate-*r/ 60.2%

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

      2. clear-num 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

      1. div-inv 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*r/ 60.0%

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

       2. *-rgt-identity 60.0%

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

       3. associate-/r* 60.0%

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

   11. Simplified 60.0%

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

       1. associate-/r/ 60.0%

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

       2. clear-num 60.2%

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

   13. Applied egg-rr 60.2%

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

   if -2.24999999999999996e-8 < b < 2.60000000000000009

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

      1. cos-sum 99.8%

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

   6. Applied egg-rr 99.8%

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

   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. associate-/l* 99.8%

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

      2. /-rgt-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. cancel-sign-sub-inv 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. fma-undefine 99.8%

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

      9. times-frac 99.8%

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

      10. *-commutative 99.8%

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

      11. times-frac 99.8%

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

      12. /-rgt-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in b around 0 98.4%

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

   if 2.60000000000000009 < b

   1. Initial program 47.9%

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

      1. associate-/l* 47.9%

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

      2. remove-double-neg 47.9%

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

      3. remove-double-neg 47.9%

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

      4. +-commutative 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*r/ 47.9%

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

      2. clear-num 48.0%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in a around 0 48.6%

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

      1. add-cube-cbrt 47.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\frac{\cos b}{r \cdot \sin b}}} \cdot \sqrt[3]{\frac{1}{\frac{\cos b}{r \cdot \sin b}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{\cos b}{r \cdot \sin b}}}} \]

      2. pow3 48.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\frac{\cos b}{r \cdot \sin b}}}\right)}^{3}} \]

      3. clear-num 48.0%

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

   9. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{r \cdot \sin b}{\cos b}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 48.5%

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

       2. clear-num 48.6%

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

       3. *-commutative 48.6%

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

       4. associate-/l/ 48.5%

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

       5. clear-num 48.6%

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

   11. Applied egg-rr 48.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 75.5%

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

   2. remove-double-neg 75.5%

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

   3. remove-double-neg 75.5%

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

   4. +-commutative 75.5%

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

3. Simplified 75.5%

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

5. Final simplification 75.5%

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

Alternative 9: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-/l* 75.6%

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

6. Applied egg-rr 75.6%

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

7. Final simplification 75.6%

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

Alternative 10: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.25e-8)
   (* r (tan b))
   (if (<= b 2.6) (* b (/ r (cos a))) (/ 1.0 (/ (/ 1.0 r) (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = r * tan(b);
	} else if (b <= 2.6) {
		tmp = b * (r / cos(a));
	} else {
		tmp = 1.0 / ((1.0 / r) / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.25d-8)) then
        tmp = r * tan(b)
    else if (b <= 2.6d0) then
        tmp = b * (r / cos(a))
    else
        tmp = 1.0d0 / ((1.0d0 / r) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.25e-8) {
		tmp = r * Math.tan(b);
	} else if (b <= 2.6) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = 1.0 / ((1.0 / r) / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.25e-8:
		tmp = r * math.tan(b)
	elif b <= 2.6:
		tmp = b * (r / math.cos(a))
	else:
		tmp = 1.0 / ((1.0 / r) / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.25e-8)
		tmp = Float64(r * tan(b));
	elseif (b <= 2.6)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.25e-8)
		tmp = r * tan(b);
	elseif (b <= 2.6)
		tmp = b * (r / cos(a));
	else
		tmp = 1.0 / ((1.0 / r) / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.25e-8], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.24999999999999996e-8

   1. Initial program 60.2%

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

      1. associate-/l* 60.1%

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

      2. remove-double-neg 60.1%

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

      3. remove-double-neg 60.1%

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

      4. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. associate-*r/ 60.2%

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

      2. clear-num 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

      1. div-inv 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*r/ 60.0%

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

       2. *-rgt-identity 60.0%

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

       3. associate-/r* 60.0%

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

   11. Simplified 60.0%

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

       1. associate-/l/ 60.0%

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

       2. *-commutative 60.0%

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

       3. clear-num 60.2%

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

       4. associate-/l* 60.1%

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

       5. quot-tan 60.2%

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

   13. Applied egg-rr 60.2%

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

   if -2.24999999999999996e-8 < b < 2.60000000000000009

   1. Initial program 98.5%

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

      1. associate-/l* 98.5%

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

      2. remove-double-neg 98.5%

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

      3. remove-double-neg 98.5%

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

      4. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in b around 0 98.4%

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

      1. associate-/l* 98.4%

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

   7. Simplified 98.4%

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

   if 2.60000000000000009 < b

   1. Initial program 47.9%

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

      1. associate-/l* 47.9%

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

      2. remove-double-neg 47.9%

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

      3. remove-double-neg 47.9%

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

      4. +-commutative 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*r/ 47.9%

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

      2. clear-num 48.0%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in a around 0 48.6%

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

      1. div-inv 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. associate-*r/ 48.6%

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

       2. *-rgt-identity 48.6%

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

       3. associate-/r* 48.5%

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

   11. Simplified 48.5%

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

       1. div-inv 48.5%

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

       2. *-un-lft-identity 48.5%

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

       3. div-inv 48.5%

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

       4. associate-/l/ 48.6%

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

       5. *-commutative 48.6%

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

       6. clear-num 48.5%

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

       7. associate-/l* 48.5%

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

       8. quot-tan 48.5%

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

   13. Applied egg-rr 48.5%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{1}{r \cdot \tan b}}} \]
   14. Step-by-step derivation

       1. *-lft-identity 48.5%

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

       2. associate-/r* 48.5%

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

   15. Simplified 48.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2.6:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8} \lor \neg \left(b \leq 2.6\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.25e-8) (not (<= b 2.6))) (* r (tan b)) (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.25e-8) || !(b <= 2.6)) {
		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 <= (-2.25d-8)) .or. (.not. (b <= 2.6d0))) 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 <= -2.25e-8) || !(b <= 2.6)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.25e-8) or not (b <= 2.6):
		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 <= -2.25e-8) || !(b <= 2.6))
		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 <= -2.25e-8) || ~((b <= 2.6)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.25e-8], N[Not[LessEqual[b, 2.6]], $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 < -2.24999999999999996e-8 or 2.60000000000000009 < b

   1. Initial program 53.9%

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

      1. associate-/l* 53.9%

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

      2. remove-double-neg 53.9%

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

      3. remove-double-neg 53.9%

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

      4. +-commutative 53.9%

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

   3. Simplified 53.9%

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

      1. associate-*r/ 53.9%

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

      2. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

   7. Taylor expanded in a around 0 54.2%

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

      1. div-inv 54.2%

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

   9. Applied egg-rr 54.2%

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

       1. associate-*r/ 54.2%

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

       2. *-rgt-identity 54.2%

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

       3. associate-/r* 54.2%

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

   11. Simplified 54.2%

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

       1. associate-/l/ 54.2%

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

       2. *-commutative 54.2%

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

       3. clear-num 54.3%

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

       4. associate-/l* 54.2%

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

       5. quot-tan 54.2%

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

   13. Applied egg-rr 54.2%

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

   if -2.24999999999999996e-8 < b < 2.60000000000000009

   1. Initial program 98.5%

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

      1. associate-/l* 98.5%

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

      2. remove-double-neg 98.5%

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

      3. remove-double-neg 98.5%

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

      4. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in b around 0 98.4%

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

      1. associate-/l* 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-8} \lor \neg \left(b \leq 2.6\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 75.5%

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

   2. remove-double-neg 75.5%

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

   3. remove-double-neg 75.5%

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

   4. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. associate-*r/ 75.5%

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

   2. clear-num 75.1%

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

6. Applied egg-rr 75.1%

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

7. Taylor expanded in a around 0 60.2%

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

   1. div-inv 60.2%

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

9. Applied egg-rr 60.2%

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

    1. associate-*r/ 60.2%

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

    2. *-rgt-identity 60.2%

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

    3. associate-/r* 60.1%

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

11. Simplified 60.1%

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

    1. associate-/l/ 60.2%

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

    2. *-commutative 60.2%

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

    3. clear-num 60.6%

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

    4. associate-/l* 60.6%

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

    5. quot-tan 60.6%

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

13. Applied egg-rr 60.6%

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

14. Final simplification 60.6%

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

Alternative 13: 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 75.5%

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

   1. associate-/l* 75.5%

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

   2. remove-double-neg 75.5%

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

   3. remove-double-neg 75.5%

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

   4. +-commutative 75.5%

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

3. Simplified 75.5%

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

5. Taylor expanded in b around 0 50.2%

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

   1. associate-/l* 50.2%

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

7. Simplified 50.2%

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

8. Taylor expanded in a around 0 35.2%

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

   1. *-commutative 35.2%

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

10. Simplified 35.2%

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

11. Final simplification 35.2%

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

Reproduce

herbie shell --seed 2024069 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))