rsin A (should all be same)

Percentage Accurate: 75.5% → 99.5%

Time: 17.3s

Alternatives: 13

Speedup: 1.0×

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, 0.2× speedup

Math

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

FPCore

(FPCore (r a b)
 :precision binary64
 (/
  (* r (sin b))
  (+
   (fma (cos b) (cos a) (* (sin b) (- (sin a))))
   (fma (- (sin b)) (sin 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))) + fma(-sin(b), sin(a), (sin(b) * sin(a))));
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / Float64(fma(cos(b), cos(a), Float64(sin(b) * Float64(-sin(a)))) + fma(Float64(-sin(b)), sin(a), Float64(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] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. prod-diff 99.5%

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

6. Applied egg-rr 99.5%

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

   1. fma-undefine 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. *-commutative 99.5%

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

   5. *-rgt-identity 99.5%

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

   6. *-commutative 99.5%

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

   7. fma-undefine 99.5%

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

   8. *-rgt-identity 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. *-rgt-identity 99.5%

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

   11. fma-undefine 99.5%

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

   12. *-commutative 99.5%

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

8. Simplified 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-define 99.5%

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \sin a\\ \frac{r \cdot \sin b}{\mathsf{fma}\left(-\sin b, \sin a, t\_0\right) + \left(\cos b \cdot \cos a - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (sin a))))
   (/
    (* r (sin b))
    (+ (fma (- (sin b)) (sin a) t_0) (- (* (cos b) (cos a)) t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * sin(a);
	return (r * sin(b)) / (fma(-sin(b), sin(a), t_0) + ((cos(b) * cos(a)) - t_0));
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * sin(a))
	return Float64(Float64(r * sin(b)) / Float64(fma(Float64(-sin(b)), sin(a), t_0) + Float64(Float64(cos(b) * cos(a)) - t_0)))
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]}, N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. prod-diff 99.5%

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

6. Applied egg-rr 99.5%

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

   1. fma-undefine 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. *-commutative 99.5%

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

   5. *-rgt-identity 99.5%

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

   6. *-commutative 99.5%

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

   7. fma-undefine 99.5%

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

   8. *-rgt-identity 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. *-rgt-identity 99.5%

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

   11. fma-undefine 99.5%

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

   12. *-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), Float64(sin(b) * Float64(-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 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.5%

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

   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. Final simplification 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]
8. 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 75.3%

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

   1. associate-/l* 75.3%

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

   2. remove-double-neg 75.3%

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

   3. remove-double-neg 75.3%

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

   4. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 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 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.4%

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

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. Final simplification 99.5%

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

Alternative 6: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00034 \lor \neg \left(a \leq 8.6 \cdot 10^{-19}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00034) (not (<= a 8.6e-19)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00034) || !(a <= 8.6e-19)) {
		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 <= (-0.00034d0)) .or. (.not. (a <= 8.6d-19))) 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 <= -0.00034) || !(a <= 8.6e-19)) {
		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 <= -0.00034) or not (a <= 8.6e-19):
		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 <= -0.00034) || !(a <= 8.6e-19))
		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 <= -0.00034) || ~((a <= 8.6e-19)))
		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, -0.00034], N[Not[LessEqual[a, 8.6e-19]], $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 < -3.4e-4 or 8.6e-19 < a

   1. Initial program 55.6%

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

      1. associate-/l* 55.6%

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

      2. remove-double-neg 55.6%

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

      3. remove-double-neg 55.6%

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

      4. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in b around 0 55.8%

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

   if -3.4e-4 < a < 8.6e-19

   1. Initial program 99.1%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. remove-double-neg 99.0%

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

      4. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in a around 0 98.2%

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

      1. rem-cube-cbrt 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. quot-tan 99.2%

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

   9. Applied egg-rr 99.2%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00034 \lor \neg \left(a \leq 8.6 \cdot 10^{-19}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00015 \lor \neg \left(a \leq 8.6 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00015) (not (<= a 8.6e-19)))
   (/ r (/ (cos a) (sin b)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00015) || !(a <= 8.6e-19)) {
		tmp = r / (cos(a) / sin(b));
	} 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 <= (-0.00015d0)) .or. (.not. (a <= 8.6d-19))) then
        tmp = r / (cos(a) / sin(b))
    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 <= -0.00015) || !(a <= 8.6e-19)) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00015) or not (a <= 8.6e-19):
		tmp = r / (math.cos(a) / math.sin(b))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00015) || !(a <= 8.6e-19))
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00015) || ~((a <= 8.6e-19)))
		tmp = r / (cos(a) / sin(b));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00015], N[Not[LessEqual[a, 8.6e-19]], $MachinePrecision]], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.49999999999999987e-4 or 8.6e-19 < a

   1. Initial program 55.6%

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

      1. associate-/l* 55.6%

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

      2. remove-double-neg 55.6%

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

      3. remove-double-neg 55.6%

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

      4. +-commutative 55.6%

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

   3. Simplified 55.6%

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

      1. clear-num 55.6%

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

      2. un-div-inv 55.6%

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

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in b around 0 55.8%

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

   if -1.49999999999999987e-4 < a < 8.6e-19

   1. Initial program 99.1%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. remove-double-neg 99.0%

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

      4. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in a around 0 98.2%

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

      1. rem-cube-cbrt 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. quot-tan 99.2%

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

   9. Applied egg-rr 99.2%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00015 \lor \neg \left(a \leq 8.6 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00045:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-19}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.00045)
   (/ (* r (sin b)) (cos a))
   (if (<= a 8.6e-19) (* r (tan b)) (/ r (/ (cos a) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00045) {
		tmp = (r * sin(b)) / cos(a);
	} else if (a <= 8.6e-19) {
		tmp = r * tan(b);
	} else {
		tmp = r / (cos(a) / sin(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.00045d0)) then
        tmp = (r * sin(b)) / cos(a)
    else if (a <= 8.6d-19) then
        tmp = r * tan(b)
    else
        tmp = r / (cos(a) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00045) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else if (a <= 8.6e-19) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r / (Math.cos(a) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.00045:
		tmp = (r * math.sin(b)) / math.cos(a)
	elif a <= 8.6e-19:
		tmp = r * math.tan(b)
	else:
		tmp = r / (math.cos(a) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.00045)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	elseif (a <= 8.6e-19)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.00045)
		tmp = (r * sin(b)) / cos(a);
	elseif (a <= 8.6e-19)
		tmp = r * tan(b);
	else
		tmp = r / (cos(a) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.00045], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.6e-19], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.4999999999999999e-4

   1. Initial program 52.8%

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

      1. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in b around 0 52.6%

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

   if -4.4999999999999999e-4 < a < 8.6e-19

   1. Initial program 99.1%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. remove-double-neg 99.0%

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

      4. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in a around 0 98.2%

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

      1. rem-cube-cbrt 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. quot-tan 99.2%

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

   9. Applied egg-rr 99.2%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   11. Simplified 99.2%

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

   if 8.6e-19 < a

   1. Initial program 57.9%

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

      1. associate-/l* 57.9%

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

      2. remove-double-neg 57.9%

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

      3. remove-double-neg 57.9%

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

      4. +-commutative 57.9%

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

   3. Simplified 57.9%

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

      1. clear-num 57.9%

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

      2. un-div-inv 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in b around 0 58.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00045:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-19}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 75.3%

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

   1. associate-/l* 75.3%

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

   2. remove-double-neg 75.3%

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

   3. remove-double-neg 75.3%

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

   4. +-commutative 75.3%

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

3. Simplified 75.3%

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

5. Final simplification 75.3%

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

Alternative 10: 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 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. *-commutative 75.3%

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

   2. associate-/l* 75.3%

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

6. Applied egg-rr 75.3%

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

7. Final simplification 75.3%

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

Alternative 11: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{-5} \lor \neg \left(b \leq 1.5 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.85e-5) (not (<= b 1.5e-7)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.85e-5) || !(b <= 1.5e-7)) {
		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 <= (-2.85d-5)) .or. (.not. (b <= 1.5d-7))) 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 <= -2.85e-5) || !(b <= 1.5e-7)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.85e-5) or not (b <= 1.5e-7):
		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 <= -2.85e-5) || !(b <= 1.5e-7))
		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 <= -2.85e-5) || ~((b <= 1.5e-7)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.85e-5], N[Not[LessEqual[b, 1.5e-7]], $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 < -2.8500000000000002e-5 or 1.4999999999999999e-7 < b

   1. Initial program 54.2%

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

      1. associate-/l* 54.1%

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

      2. remove-double-neg 54.1%

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

      3. remove-double-neg 54.1%

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

      4. +-commutative 54.1%

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

   3. Simplified 54.1%

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

      1. add-cube-cbrt 53.3%

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

      2. pow3 53.3%

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

   6. Applied egg-rr 53.3%

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

   7. Taylor expanded in a around 0 52.5%

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

      1. rem-cube-cbrt 53.3%

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

      2. *-un-lft-identity 53.3%

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

      3. quot-tan 53.4%

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

   9. Applied egg-rr 53.4%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 53.4%

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

   11. Simplified 53.4%

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

   if -2.8500000000000002e-5 < b < 1.4999999999999999e-7

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   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. *-un-lft-identity 99.7%

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

      3. prod-diff 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. fma-undefine 99.7%

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

      2. distribute-lft-neg-in 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-rgt-identity 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. fma-undefine 99.7%

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

      12. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in b around 0 99.7%

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

       1. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{-5} \lor \neg \left(b \leq 1.5 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% 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 75.3%

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

   1. associate-/l* 75.3%

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

   2. remove-double-neg 75.3%

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

   3. remove-double-neg 75.3%

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

   4. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. add-cube-cbrt 74.6%

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

   2. pow3 74.6%

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

6. Applied egg-rr 74.6%

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

7. Taylor expanded in a around 0 56.6%

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

   1. rem-cube-cbrt 57.0%

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

   2. *-un-lft-identity 57.0%

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

   3. quot-tan 57.1%

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

9. Applied egg-rr 57.1%

   \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 57.1%

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

11. Simplified 57.1%

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

12. Final simplification 57.1%

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

Alternative 13: 34.4% 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 75.3%

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

   1. associate-/l* 75.3%

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

   2. remove-double-neg 75.3%

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

   3. remove-double-neg 75.3%

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

   4. +-commutative 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in b around 0 49.4%

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

6. Taylor expanded in a around 0 31.4%

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

7. Final simplification 31.4%

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

Reproduce

herbie shell --seed 2024068 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))