rsin B (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 16.8s

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

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 75.1%

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

   1. associate-*r/ 75.1%

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

   2. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 3: 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 75.1%

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

   1. associate-*r/ 75.1%

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

   2. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 4: 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 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

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. Add Preprocessing

Alternative 5: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (cos b) (cos a) 0.0))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / fma(cos(b), cos(a), 0.0));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), 0.0)))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 52.1%

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

   2. sqrt-unprod 85.3%

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

   3. sqr-neg 85.3%

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

   4. sqrt-unprod 33.2%

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

   5. add-sqr-sqrt 74.8%

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

   6. sin-mult 76.0%

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

   7. cos-sum 74.3%

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

   8. cancel-sign-sub-inv 74.3%

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

   9. add-sqr-sqrt 41.2%

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

   10. sqrt-unprod 75.2%

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

   11. sqr-neg 75.2%

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

   12. sqrt-unprod 34.0%

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

   13. add-sqr-sqrt 76.0%

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

   14. cos-diff 76.0%

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

8. Applied egg-rr 76.0%

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

   1. +-inverses 76.0%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\color{blue}{0}}{2}\right)} \]

   2. metadata-eval 76.0%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)} \]

10. Simplified 76.0%

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)} \]
11. Add Preprocessing

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -75000000000000 \lor \neg \left(a \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -75000000000000.0) (not (<= a 2.9e-8)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -75000000000000.0) || !(a <= 2.9e-8)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = 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 ((a <= (-75000000000000.0d0)) .or. (.not. (a <= 2.9d-8))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -75000000000000.0) || !(a <= 2.9e-8)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -75000000000000.0) or not (a <= 2.9e-8):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -75000000000000.0) || !(a <= 2.9e-8))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -75000000000000.0) || ~((a <= 2.9e-8)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -75000000000000.0], N[Not[LessEqual[a, 2.9e-8]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e13 or 2.9000000000000002e-8 < a

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in b around 0 57.9%

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

   if -7.5e13 < a < 2.9000000000000002e-8

   1. Initial program 97.4%

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

      1. associate-*r/ 97.3%

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

      2. +-commutative 97.3%

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

   3. Simplified 97.3%

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

      1. cos-sum 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. add-sqr-sqrt 61.3%

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

      5. sqrt-unprod 60.9%

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

      6. sqr-neg 60.9%

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

      7. sqrt-unprod 6.1%

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

      8. add-sqr-sqrt 18.6%

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

      9. distribute-lft-neg-out 18.6%

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

      10. fma-neg 18.6%

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

      11. cos-sum 18.8%

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

      12. cos-sum 18.6%

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

      13. cancel-sign-sub-inv 18.6%

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

   8. Applied egg-rr 18.8%

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

      1. associate-*r/ 18.8%

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

      2. *-rgt-identity 18.8%

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

      3. distribute-frac-neg2 18.8%

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

      4. associate-*l/ 18.8%

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

      5. associate-/r/ 18.8%

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

      6. distribute-neg-frac 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in a around 0 18.8%

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

       1. mul-1-neg 18.8%

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

       2. associate-/l* 18.8%

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

       3. distribute-rgt-neg-in 18.8%

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

   13. Simplified 18.8%

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

       1. add-sqr-sqrt 17.3%

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

       2. sqrt-unprod 43.1%

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

       3. distribute-rgt-neg-out 43.1%

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

       4. distribute-rgt-neg-out 43.1%

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

       5. sqr-neg 43.1%

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

       6. sqrt-unprod 54.4%

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

       7. add-sqr-sqrt 97.4%

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

       8. clear-num 97.2%

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

       9. un-div-inv 97.2%

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

       10. clear-num 97.2%

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

       11. quot-tan 97.2%

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

   15. Applied egg-rr 97.2%

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

       1. associate-/r/ 97.5%

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

       2. /-rgt-identity 97.5%

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

   17. Simplified 97.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -75000000000000 \lor \neg \left(a \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -75000000000000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -75000000000000.0)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2.9e-8) (* r (tan b)) (/ (* r (sin b)) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -75000000000000.0) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2.9e-8) {
		tmp = r * tan(b);
	} else {
		tmp = (r * sin(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 (a <= (-75000000000000.0d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2.9d-8) then
        tmp = r * tan(b)
    else
        tmp = (r * sin(b)) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -75000000000000.0) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2.9e-8) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -75000000000000.0:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2.9e-8:
		tmp = r * math.tan(b)
	else:
		tmp = (r * math.sin(b)) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -75000000000000.0)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2.9e-8)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -75000000000000.0)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2.9e-8)
		tmp = r * tan(b);
	else
		tmp = (r * sin(b)) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -75000000000000.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.9e-8], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.5e13

   1. Initial program 52.9%

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

      1. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in b around 0 52.5%

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

   if -7.5e13 < a < 2.9000000000000002e-8

   1. Initial program 97.4%

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

      1. associate-*r/ 97.3%

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

      2. +-commutative 97.3%

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

   3. Simplified 97.3%

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

      1. cos-sum 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. add-sqr-sqrt 61.3%

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

      5. sqrt-unprod 60.9%

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

      6. sqr-neg 60.9%

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

      7. sqrt-unprod 6.1%

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

      8. add-sqr-sqrt 18.6%

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

      9. distribute-lft-neg-out 18.6%

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

      10. fma-neg 18.6%

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

      11. cos-sum 18.8%

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

      12. cos-sum 18.6%

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

      13. cancel-sign-sub-inv 18.6%

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

   8. Applied egg-rr 18.8%

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

      1. associate-*r/ 18.8%

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

      2. *-rgt-identity 18.8%

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

      3. distribute-frac-neg2 18.8%

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

      4. associate-*l/ 18.8%

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

      5. associate-/r/ 18.8%

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

      6. distribute-neg-frac 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in a around 0 18.8%

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

       1. mul-1-neg 18.8%

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

       2. associate-/l* 18.8%

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

       3. distribute-rgt-neg-in 18.8%

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

   13. Simplified 18.8%

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

       1. add-sqr-sqrt 17.3%

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

       2. sqrt-unprod 43.1%

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

       3. distribute-rgt-neg-out 43.1%

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

       4. distribute-rgt-neg-out 43.1%

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

       5. sqr-neg 43.1%

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

       6. sqrt-unprod 54.4%

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

       7. add-sqr-sqrt 97.4%

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

       8. clear-num 97.2%

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

       9. un-div-inv 97.2%

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

       10. clear-num 97.2%

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

       11. quot-tan 97.2%

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

   15. Applied egg-rr 97.2%

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

       1. associate-/r/ 97.5%

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

       2. /-rgt-identity 97.5%

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

   17. Simplified 97.5%

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

   if 2.9000000000000002e-8 < a

   1. Initial program 64.7%

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

      1. associate-*r/ 64.7%

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

      2. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in b around 0 64.3%

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

Alternative 8: 76.9% 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 75.1%

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

3. Final simplification 75.1%

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

Alternative 9: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0015 \lor \neg \left(b \leq 0.00013\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0015) (not (<= b 0.00013)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0015) || !(b <= 0.00013)) {
		tmp = r * tan(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.0015d0)) .or. (.not. (b <= 0.00013d0))) then
        tmp = r * tan(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0015) || !(b <= 0.00013)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0015) or not (b <= 0.00013):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0015) || !(b <= 0.00013))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0015) || ~((b <= 0.00013)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0015], N[Not[LessEqual[b, 0.00013]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0015 or 1.29999999999999989e-4 < b

   1. Initial program 49.4%

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

      1. associate-*r/ 49.4%

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

      2. +-commutative 49.4%

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

   3. Simplified 49.4%

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

      1. cos-sum 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. fma-define 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. div-inv 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

      4. add-sqr-sqrt 50.2%

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

      5. sqrt-unprod 51.3%

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

      6. sqr-neg 51.3%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 2.0%

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

      9. distribute-lft-neg-out 2.0%

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

      10. fma-neg 2.0%

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

      11. cos-sum 6.5%

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

      12. cos-sum 2.0%

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

      13. cancel-sign-sub-inv 2.0%

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

   8. Applied egg-rr 7.1%

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

      1. associate-*r/ 7.1%

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

      2. *-rgt-identity 7.1%

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

      3. distribute-frac-neg2 7.1%

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

      4. associate-*l/ 7.1%

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

      5. associate-/r/ 7.1%

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

      6. distribute-neg-frac 7.1%

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

   10. Simplified 7.1%

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

   11. Taylor expanded in a around 0 7.2%

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

       1. mul-1-neg 7.2%

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

       2. associate-/l* 7.2%

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

       3. distribute-rgt-neg-in 7.2%

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

   13. Simplified 7.2%

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

       1. add-sqr-sqrt 3.8%

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

       2. sqrt-unprod 17.3%

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

       3. distribute-rgt-neg-out 17.3%

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

       4. distribute-rgt-neg-out 17.3%

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

       5. sqr-neg 17.3%

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

       6. sqrt-unprod 23.3%

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

       7. add-sqr-sqrt 48.7%

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

       8. clear-num 48.5%

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

       9. un-div-inv 48.6%

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

       10. clear-num 48.6%

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

       11. quot-tan 48.7%

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

   15. Applied egg-rr 48.7%

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

       1. associate-/r/ 48.7%

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

       2. /-rgt-identity 48.7%

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

   17. Simplified 48.7%

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

   if -0.0015 < b < 1.29999999999999989e-4

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

   3. Simplified 99.2%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0 99.2%

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

      1. cos-neg 99.2%

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

      2. associate-/l* 99.2%

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

      3. cos-neg 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0015 \lor \neg \left(b \leq 0.00013\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.6% 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 75.1%

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

   1. associate-*r/ 75.1%

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

   2. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

   1. frac-2neg 99.5%

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

   2. div-inv 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. add-sqr-sqrt 51.9%

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

   5. sqrt-unprod 55.5%

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

   6. sqr-neg 55.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 20.0%

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

   9. distribute-lft-neg-out 20.0%

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

   10. fma-neg 20.0%

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

   11. cos-sum 22.1%

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

   12. cos-sum 20.0%

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

   13. cancel-sign-sub-inv 20.0%

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

8. Applied egg-rr 22.5%

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

   1. associate-*r/ 22.5%

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

   2. *-rgt-identity 22.5%

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

   3. distribute-frac-neg2 22.5%

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

   4. associate-*l/ 22.5%

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

   5. associate-/r/ 22.5%

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

   6. distribute-neg-frac 22.5%

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

10. Simplified 22.5%

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

11. Taylor expanded in a around 0 24.1%

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

    1. mul-1-neg 24.1%

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

    2. associate-/l* 24.1%

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

    3. distribute-rgt-neg-in 24.1%

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

13. Simplified 24.1%

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

    1. add-sqr-sqrt 20.1%

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

    2. sqrt-unprod 33.0%

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

    3. distribute-rgt-neg-out 33.0%

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

    4. distribute-rgt-neg-out 33.0%

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

    5. sqr-neg 33.0%

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

    6. sqrt-unprod 36.8%

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

    7. add-sqr-sqrt 57.9%

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

    8. clear-num 57.9%

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

    9. un-div-inv 57.9%

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

    10. clear-num 57.9%

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

    11. quot-tan 57.9%

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

15. Applied egg-rr 57.9%

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

    1. associate-/r/ 58.0%

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

    2. /-rgt-identity 58.0%

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

17. Simplified 58.0%

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

Alternative 11: 35.2% 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 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in b around 0 52.9%

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

6. Taylor expanded in a around 0 36.2%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))