rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 16.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

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 77.0%

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

   1. associate-*r/ 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-*r/ 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in r around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-lft-neg-out 99.6%

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

   3. +-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. cancel-sign-sub-inv 99.6%

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

   6. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. sub-neg 99.6%

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

10. Applied egg-rr 99.6%

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

    1. +-commutative 99.6%

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

    2. distribute-lft-neg-in 99.6%

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

    3. *-commutative 99.6%

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

    4. fma-def 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in b around inf 99.6%

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

    1. *-commutative 99.6%

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

    2. associate-*r* 99.6%

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

    3. neg-mul-1 99.6%

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

    4. *-commutative 99.6%

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

    5. fma-def 99.6%

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

    6. associate-/l* 99.5%

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

    7. fma-def 99.5%

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

    8. +-commutative 99.5%

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

    9. distribute-rgt-neg-out 99.5%

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

    10. *-commutative 99.5%

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

    11. unsub-neg 99.5%

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

15. Simplified 99.5%

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

16. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-*r/ 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00052 \lor \neg \left(a \leq 2050000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00052) (not (<= a 2050000000000.0)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00052) || !(a <= 2050000000000.0)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.00052d0)) .or. (.not. (a <= 2050000000000.0d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00052) || !(a <= 2050000000000.0)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00052) or not (a <= 2050000000000.0):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00052) || !(a <= 2050000000000.0))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00052) || ~((a <= 2050000000000.0)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00052], N[Not[LessEqual[a, 2050000000000.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.19999999999999954e-4 or 2.05e12 < a

   1. Initial program 58.9%

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

      1. +-commutative 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in b around 0 58.5%

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

   if -5.19999999999999954e-4 < a < 2.05e12

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00052 \lor \neg \left(a \leq 2050000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000195:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2050000000000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000195)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2050000000000.0)
     (* r (/ (sin b) (cos b)))
     (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000195) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2050000000000.0) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = sin(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 (a <= (-0.000195d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2050000000000.0d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000195) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2050000000000.0) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000195:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2050000000000.0:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000195)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2050000000000.0)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000195)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2050000000000.0)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000195], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2050000000000.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.94999999999999996e-4

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 57.6%

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

   if -1.94999999999999996e-4 < a < 2.05e12

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.1%

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

   if 2.05e12 < a

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-/r/ 59.3%

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

      3. +-commutative 59.3%

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

   3. Simplified 59.3%

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

      1. clear-num 59.3%

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

      2. inv-pow 59.3%

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

   5. Applied egg-rr 59.3%

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

      1. unpow-1 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in b around 0 59.3%

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

   9. Taylor expanded in b around inf 59.4%

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

       1. associate-*r/ 59.4%

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

       2. *-commutative 59.4%

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

   11. Simplified 59.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000195:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2050000000000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.002:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2050000000000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.002)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2050000000000.0)
     (* r (/ (sin b) (cos b)))
     (/ r (/ (cos a) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.002) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2050000000000.0) {
		tmp = r * (sin(b) / cos(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 <= (-0.002d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2050000000000.0d0) then
        tmp = r * (sin(b) / cos(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 <= -0.002) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2050000000000.0) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r / (Math.cos(a) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.002:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2050000000000.0:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r / (math.cos(a) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.002)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2050000000000.0)
		tmp = Float64(r * Float64(sin(b) / cos(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 <= -0.002)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2050000000000.0)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r / (cos(a) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.002], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2050000000000.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $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 < -2e-3

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 57.6%

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

   if -2e-3 < a < 2.05e12

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.1%

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

   if 2.05e12 < a

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-/r/ 59.3%

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

      3. +-commutative 59.3%

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

   3. Simplified 59.3%

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

      1. clear-num 59.3%

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

      2. inv-pow 59.3%

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

   5. Applied egg-rr 59.3%

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

      1. unpow-1 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in b around 0 59.3%

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

   9. Taylor expanded in b around inf 59.4%

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

       1. *-commutative 59.4%

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

       2. associate-/l* 59.5%

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

   11. Simplified 59.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.002:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2050000000000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 8: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2050000000000:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -1.15e-5)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2050000000000.0)
     (/ (* r (sin b)) (cos b))
     (/ r (/ (cos a) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -1.15e-5) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2050000000000.0) {
		tmp = (r * sin(b)) / cos(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 <= (-1.15d-5)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2050000000000.0d0) then
        tmp = (r * sin(b)) / cos(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 <= -1.15e-5) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2050000000000.0) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else {
		tmp = r / (Math.cos(a) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -1.15e-5:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2050000000000.0:
		tmp = (r * math.sin(b)) / math.cos(b)
	else:
		tmp = r / (math.cos(a) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -1.15e-5)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2050000000000.0)
		tmp = Float64(Float64(r * sin(b)) / cos(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 <= -1.15e-5)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2050000000000.0)
		tmp = (r * sin(b)) / cos(b);
	else
		tmp = r / (cos(a) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -1.15e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2050000000000.0], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[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 < -1.15e-5

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 57.6%

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

   if -1.15e-5 < a < 2.05e12

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.2%

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

   if 2.05e12 < a

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-/r/ 59.3%

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

      3. +-commutative 59.3%

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

   3. Simplified 59.3%

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

      1. clear-num 59.3%

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

      2. inv-pow 59.3%

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

   5. Applied egg-rr 59.3%

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

      1. unpow-1 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in b around 0 59.3%

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

   9. Taylor expanded in b around inf 59.4%

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

       1. *-commutative 59.4%

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

       2. associate-/l* 59.5%

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

   11. Simplified 59.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2050000000000:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 9: 76.8% 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 77.0%

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

2. Final simplification 77.0%

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

Alternative 10: 76.8% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 77.0%

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

   1. associate-*r/ 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Final simplification 77.0%

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

Alternative 11: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 55.6%

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

5. Final simplification 55.6%

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

Alternative 12: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 52.2%

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

5. Final simplification 52.2%

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

Alternative 13: 39.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. *-commutative 77.0%

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

   2. associate-/r/ 76.9%

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

   3. +-commutative 76.9%

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

3. Simplified 76.9%

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

   1. clear-num 76.8%

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

   2. inv-pow 76.8%

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

5. Applied egg-rr 76.8%

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

   1. unpow-1 76.8%

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

   2. +-commutative 76.8%

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

7. Simplified 76.8%

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

8. Taylor expanded in b around 0 55.4%

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

9. Taylor expanded in a around 0 37.3%

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

10. Final simplification 37.3%

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

Alternative 14: 35.8% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{1}{b \cdot -0.3333333333333333 + \frac{1}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ 1.0 (+ (* b -0.3333333333333333) (/ 1.0 b)))))

C

double code(double r, double a, double b) {
	return r * (1.0 / ((b * -0.3333333333333333) + (1.0 / 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 * (1.0d0 / ((b * (-0.3333333333333333d0)) + (1.0d0 / b)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (1.0 / ((b * -0.3333333333333333) + (1.0 / b)));
}

Python

def code(r, a, b):
	return r * (1.0 / ((b * -0.3333333333333333) + (1.0 / b)))

Julia

function code(r, a, b)
	return Float64(r * Float64(1.0 / Float64(Float64(b * -0.3333333333333333) + Float64(1.0 / b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (1.0 / ((b * -0.3333333333333333) + (1.0 / b)));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(1.0 / N[(N[(b * -0.3333333333333333), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

   1. associate-*r/ 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-/l* 99.5%

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

   2. div-inv 99.5%

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

   3. distribute-lft-neg-out 99.5%

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

   4. fma-neg 99.5%

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

   5. cos-sum 76.9%

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

7. Applied egg-rr 76.9%

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

8. Taylor expanded in b around 0 52.8%

   \[\leadsto r \cdot \frac{1}{\color{blue}{b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \left(-1 \cdot \sin a + \frac{\cos a}{b}\right)}} \]
9. Step-by-step derivation

   1. fma-def 52.8%

      \[\leadsto r \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, -0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a, -1 \cdot \sin a + \frac{\cos a}{b}\right)}} \]

   2. distribute-rgt-out-- 52.8%

      \[\leadsto r \cdot \frac{1}{\mathsf{fma}\left(b, \color{blue}{\cos a \cdot \left(-0.5 - -0.16666666666666666\right)}, -1 \cdot \sin a + \frac{\cos a}{b}\right)} \]

   3. metadata-eval 52.8%

      \[\leadsto r \cdot \frac{1}{\mathsf{fma}\left(b, \cos a \cdot \color{blue}{-0.3333333333333333}, -1 \cdot \sin a + \frac{\cos a}{b}\right)} \]

   4. neg-mul-1 52.8%

      \[\leadsto r \cdot \frac{1}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\left(-\sin a\right)} + \frac{\cos a}{b}\right)} \]

   5. +-commutative 52.8%

      \[\leadsto r \cdot \frac{1}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\frac{\cos a}{b} + \left(-\sin a\right)}\right)} \]

   6. unsub-neg 52.8%

      \[\leadsto r \cdot \frac{1}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\frac{\cos a}{b} - \sin a}\right)} \]

10. Simplified 52.8%

    \[\leadsto r \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \frac{\cos a}{b} - \sin a\right)}} \]

11. Taylor expanded in a around 0 34.1%

    \[\leadsto r \cdot \color{blue}{\frac{1}{-0.3333333333333333 \cdot b + \frac{1}{b}}} \]

12. Final simplification 34.1%

    \[\leadsto r \cdot \frac{1}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]

Alternative 15: 35.3% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(r, a, b):
	return r * b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 52.2%

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

5. Taylor expanded in a around 0 34.1%

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

6. Final simplification 34.1%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023207 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))