rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 24.0s

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: 76.3% 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

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

FPCore

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

C

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

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * sin(a))
	return Float64(Float64(r * sin(b)) / Float64(Float64(Float64(cos(a) * cos(b)) - t_0) + fma(Float64(-sin(b)), sin(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[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.5%

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

   1. +-commutative 73.5%

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

3. Simplified 73.5%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. prod-diff 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. fma-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-undefine 99.4%

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

   6. *-rgt-identity 99.4%

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

   7. distribute-lft-neg-in 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. fma-undefine 99.4%

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

   10. *-commutative 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. associate-/l* 73.6%

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

   2. remove-double-neg 73.6%

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

   3. remove-double-neg 73.6%

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

   4. +-commutative 73.6%

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

3. Simplified 73.6%

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

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. +-commutative 73.5%

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

3. Simplified 73.5%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 77.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. +-commutative 73.5%

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

3. Simplified 73.5%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-define 99.4%

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

6. Applied egg-rr 99.4%

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

8. Applied egg-rr 12.6%

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

   1. count-2 12.6%

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

   2. *-commutative 12.6%

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

10. Simplified 12.6%

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

12. Applied egg-rr 75.5%

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

    1. fma-undefine 75.5%

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

    2. *-commutative 75.5%

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

    3. fma-define 75.5%

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

14. Simplified 75.5%

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

15. Final simplification 75.5%

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

Alternative 5: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. +-commutative 73.5%

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

3. Simplified 73.5%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-define 99.4%

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

6. Applied egg-rr 99.4%

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

8. Applied egg-rr 74.6%

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

9. Final simplification 74.6%

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

Alternative 6: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. associate-/l* 73.6%

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

   2. remove-double-neg 73.6%

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

   3. remove-double-neg 73.6%

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

   4. +-commutative 73.6%

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

3. Simplified 73.6%

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

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

8. Applied egg-rr 74.1%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\left(\mathsf{fma}\left(1, \cos \left(b + a\right), -\sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)}} \]
9. Step-by-step derivation

   1. fma-undefine 74.1%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \left(\color{blue}{\left(1 \cdot \cos \left(b + a\right) + \left(-\sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} \]

   2. *-lft-identity 74.1%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \left(\left(\color{blue}{\cos \left(b + a\right)} + \left(-\sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} \]

   3. unpow2 74.1%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \left(\left(\cos \left(b + a\right) + \left(-\sqrt[3]{\cos \left(b + a\right)} \cdot \color{blue}{\left(\sqrt[3]{\cos \left(b + a\right)} \cdot \sqrt[3]{\cos \left(b + a\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} \]

   4. rem-3cbrt-rft 74.7%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \left(\left(\cos \left(b + a\right) + \left(-\color{blue}{\cos \left(b + a\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} \]

   5. sub-neg 74.7%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \left(\color{blue}{\left(\cos \left(b + a\right) - \cos \left(b + a\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} \]

   6. +-inverses 74.7%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \left(\color{blue}{0} + \mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)\right)} \]

   7. +-lft-identity 74.7%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\cos \left(b + a\right)}, {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}, \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)}} \]

   8. fma-undefine 74.7%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\left(\left(-\sqrt[3]{\cos \left(b + a\right)}\right) \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2} + \sqrt[3]{\cos \left(b + a\right)} \cdot {\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2}\right)}} \]

   9. distribute-rgt-out 74.7%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{{\left(\sqrt[3]{\cos \left(b + a\right)}\right)}^{2} \cdot \left(\left(-\sqrt[3]{\cos \left(b + a\right)}\right) + \sqrt[3]{\cos \left(b + a\right)}\right)}} \]

10. Simplified 74.7%

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

11. Final simplification 74.7%

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

Alternative 7: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-5} \lor \neg \left(a \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -4e-5) (not (<= a 1.2e-7)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -4e-5) || !(a <= 1.2e-7)) {
		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 <= (-4d-5)) .or. (.not. (a <= 1.2d-7))) 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 <= -4e-5) || !(a <= 1.2e-7)) {
		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 <= -4e-5) or not (a <= 1.2e-7):
		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 <= -4e-5) || !(a <= 1.2e-7))
		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 <= -4e-5) || ~((a <= 1.2e-7)))
		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, -4e-5], N[Not[LessEqual[a, 1.2e-7]], $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 < -4.00000000000000033e-5 or 1.19999999999999989e-7 < a

   1. Initial program 51.5%

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

      1. associate-/l* 51.6%

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

      2. remove-double-neg 51.6%

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

      3. remove-double-neg 51.6%

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

      4. +-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in b around 0 52.2%

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

   if -4.00000000000000033e-5 < a < 1.19999999999999989e-7

   1. Initial program 98.1%

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

      1. +-commutative 98.1%

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

   3. Simplified 98.1%

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

      1. associate-*r/ 98.0%

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

      2. *-commutative 98.0%

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

      3. add-sqr-sqrt 54.2%

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

      4. associate-*r* 54.3%

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

   6. Applied egg-rr 54.3%

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

   7. Taylor expanded in a around 0 54.3%

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

      1. pow1 54.3%

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

      2. *-commutative 54.3%

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

      3. *-commutative 54.3%

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

      4. associate-*r* 54.2%

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

      5. add-sqr-sqrt 98.0%

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

      6. quot-tan 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. unpow1 98.2%

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

   11. Simplified 98.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-5} \lor \neg \left(a \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% 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 73.5%

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

   1. associate-/l* 73.6%

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

   2. remove-double-neg 73.6%

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

   3. remove-double-neg 73.6%

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

   4. +-commutative 73.6%

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

3. Simplified 73.6%

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

5. Final simplification 73.6%

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

Alternative 9: 76.3% 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 73.5%

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

   1. +-commutative 73.5%

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

3. Simplified 73.5%

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

   1. *-commutative 73.5%

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

   2. associate-/l* 73.6%

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

6. Applied egg-rr 73.6%

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

7. Final simplification 73.6%

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

Alternative 10: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.028 \lor \neg \left(b \leq 0.00058\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.028) (not (<= b 0.00058)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.028) || !(b <= 0.00058)) {
		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 <= (-0.028d0)) .or. (.not. (b <= 0.00058d0))) 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 <= -0.028) || !(b <= 0.00058)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.028) or not (b <= 0.00058):
		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 <= -0.028) || !(b <= 0.00058))
		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 <= -0.028) || ~((b <= 0.00058)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.028], N[Not[LessEqual[b, 0.00058]], $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 < -0.0280000000000000006 or 5.8e-4 < b

   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. Step-by-step derivation

      1. associate-*r/ 52.8%

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

      2. *-commutative 52.8%

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

      3. add-sqr-sqrt 29.5%

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

      4. associate-*r* 29.5%

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

   6. Applied egg-rr 29.5%

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

   7. Taylor expanded in a around 0 29.1%

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

      1. pow1 29.1%

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

      2. *-commutative 29.1%

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

      3. *-commutative 29.1%

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

      4. associate-*r* 29.1%

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

      5. add-sqr-sqrt 52.3%

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

      6. quot-tan 52.4%

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

   9. Applied egg-rr 52.4%

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

       1. unpow1 52.4%

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

   11. Simplified 52.4%

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

   if -0.0280000000000000006 < b < 5.8e-4

   1. Initial program 98.5%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. remove-double-neg 98.7%

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

      4. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in b around 0 98.5%

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

      1. associate-/l* 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.028 \lor \neg \left(b \leq 0.00058\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-5} \lor \neg \left(b \leq 0.0006\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5.5e-5) (not (<= b 0.0006)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5.5e-5) || !(b <= 0.0006)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-5.5d-5)) .or. (.not. (b <= 0.0006d0))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5.5e-5) || !(b <= 0.0006)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -5.5e-5) or not (b <= 0.0006):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -5.5e-5) || !(b <= 0.0006))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -5.5e-5) || ~((b <= 0.0006)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -5.5e-5], N[Not[LessEqual[b, 0.0006]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.5000000000000002e-5 or 5.99999999999999947e-4 < b

   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. Step-by-step derivation

      1. associate-*r/ 52.8%

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

      2. *-commutative 52.8%

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

      3. add-sqr-sqrt 29.5%

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

      4. associate-*r* 29.5%

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

   6. Applied egg-rr 29.5%

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

   7. Taylor expanded in a around 0 29.1%

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

      1. pow1 29.1%

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

      2. *-commutative 29.1%

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

      3. *-commutative 29.1%

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

      4. associate-*r* 29.1%

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

      5. add-sqr-sqrt 52.3%

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

      6. quot-tan 52.4%

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

   9. Applied egg-rr 52.4%

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

       1. unpow1 52.4%

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

   11. Simplified 52.4%

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

   if -5.5000000000000002e-5 < b < 5.99999999999999947e-4

   1. Initial program 98.5%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. remove-double-neg 98.7%

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

      4. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in b around 0 98.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-5} \lor \neg \left(b \leq 0.0006\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.4% 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 73.5%

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

   1. +-commutative 73.5%

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

3. Simplified 73.5%

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

   1. associate-*r/ 73.6%

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

   2. *-commutative 73.6%

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

   3. add-sqr-sqrt 38.6%

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

   4. associate-*r* 38.6%

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

6. Applied egg-rr 38.6%

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

7. Taylor expanded in a around 0 31.6%

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

   1. pow1 31.6%

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

   2. *-commutative 31.6%

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

   3. *-commutative 31.6%

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

   4. associate-*r* 31.6%

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

   5. add-sqr-sqrt 58.9%

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

   6. quot-tan 59.0%

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

9. Applied egg-rr 59.0%

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

    1. unpow1 59.0%

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

11. Simplified 59.0%

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

12. Final simplification 59.0%

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

Alternative 13: 33.7% 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 73.5%

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

   1. associate-/l* 73.6%

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

   2. remove-double-neg 73.6%

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

   3. remove-double-neg 73.6%

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

   4. +-commutative 73.6%

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

3. Simplified 73.6%

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

5. Taylor expanded in b around 0 47.0%

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

   1. associate-/l* 47.0%

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

7. Simplified 47.0%

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

8. Taylor expanded in a around 0 32.4%

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

9. Final simplification 32.4%

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

Reproduce

herbie shell --seed 2024044 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))