rsin B (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 15.0s

Alternatives: 10

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: 76.3% 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, 0.4× 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 80.1%

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

   1. associate-*r/ 80.2%

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

   2. +-commutative 80.2%

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

3. Simplified 80.2%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. +-commutative 80.1%

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

3. Simplified 80.1%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.5%

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

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2e-6)
   (* r (tan b))
   (if (<= b 6.8e-9) (* b (/ r (cos a))) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2e-6) {
		tmp = r * tan(b);
	} else if (b <= 6.8e-9) {
		tmp = b * (r / 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 (b <= (-2d-6)) then
        tmp = r * tan(b)
    else if (b <= 6.8d-9) then
        tmp = b * (r / 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 (b <= -2e-6) {
		tmp = r * Math.tan(b);
	} else if (b <= 6.8e-9) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2e-6:
		tmp = r * math.tan(b)
	elif b <= 6.8e-9:
		tmp = b * (r / math.cos(a))
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2e-6)
		tmp = Float64(r * tan(b));
	elseif (b <= 6.8e-9)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2e-6)
		tmp = r * tan(b);
	elseif (b <= 6.8e-9)
		tmp = b * (r / cos(a));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2e-6], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-9], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999991e-6

   1. Initial program 53.6%

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

      1. +-commutative 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in a around 0 54.5%

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

      1. associate-/l* 54.4%

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

      2. *-commutative 54.4%

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

      3. quot-tan 54.7%

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

   7. Applied egg-rr 54.7%

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

   if -1.99999999999999991e-6 < b < 6.7999999999999997e-9

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in b around 0 99.3%

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

      1. associate-/l* 99.3%

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

   7. Simplified 99.3%

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

   if 6.7999999999999997e-9 < b

   1. Initial program 65.2%

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

      1. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in a around 0 65.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% 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 80.1%

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

   1. +-commutative 80.1%

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

3. Simplified 80.1%

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

   1. cos-sum 99.5%

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

   2. sub-neg 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. fma-define 99.6%

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

   4. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. expm1-log1p-u 99.6%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b\right)\right)}, -\sin a, \cos a \cdot \cos b\right)} \]

   2. expm1-undefine 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(\sin b\right)} - 1}, -\sin a, \cos a \cdot \cos b\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(\sin b\right)} - 1}, -\sin a, \cos a \cdot \cos b\right)} \]
11. Step-by-step derivation

    1. expm1-define 99.6%

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b\right)\right)}, -\sin a, \cos a \cdot \cos b\right)} \]

12. Simplified 99.6%

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b\right)\right)}, -\sin a, \cos a \cdot \cos b\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 99.6%

       \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b\right)\right), -\sin a, \cos a \cdot \cos b\right)}} \]

    2. expm1-log1p-u 99.6%

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

    3. fma-undefine 99.6%

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

    4. *-commutative 99.6%

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

    5. +-commutative 99.6%

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

    6. add-sqr-sqrt 52.9%

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

    7. sqrt-unprod 88.6%

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

    8. sqr-neg 88.6%

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

    9. sqrt-unprod 35.7%

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

    10. add-sqr-sqrt 80.2%

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

    11. cos-diff 80.4%

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

14. Applied egg-rr 80.4%

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

Alternative 5: 76.2% 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]

Derivation

1. Initial program 80.1%

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

   1. +-commutative 80.1%

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

3. Simplified 80.1%

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

   1. cos-sum 99.5%

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

   2. sub-neg 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. fma-define 99.6%

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

   4. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. fma-undefine 99.5%

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

   2. +-commutative 99.5%

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

   3. *-commutative 99.5%

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

   4. add-sqr-sqrt 52.9%

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

   5. sqrt-unprod 88.6%

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

   6. sqr-neg 88.6%

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

   7. sqrt-unprod 35.7%

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

   8. add-sqr-sqrt 80.2%

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

   9. cos-diff 80.3%

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

10. Applied egg-rr 80.3%

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

Alternative 6: 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 80.1%

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

3. Final simplification 80.1%

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

Alternative 7: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6} \lor \neg \left(b \leq 6.8 \cdot 10^{-9}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.1e-6) (not (<= b 6.8e-9)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.1e-6) || !(b <= 6.8e-9)) {
		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 <= (-1.1d-6)) .or. (.not. (b <= 6.8d-9))) 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 <= -1.1e-6) || !(b <= 6.8e-9)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.1e-6) or not (b <= 6.8e-9):
		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 <= -1.1e-6) || !(b <= 6.8e-9))
		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 <= -1.1e-6) || ~((b <= 6.8e-9)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.1e-6], N[Not[LessEqual[b, 6.8e-9]], $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 < -1.1000000000000001e-6 or 6.7999999999999997e-9 < b

   1. Initial program 59.4%

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

      1. +-commutative 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in a around 0 60.2%

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

      1. associate-/l* 60.1%

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

      2. *-commutative 60.1%

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

      3. quot-tan 60.3%

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

   7. Applied egg-rr 60.3%

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

   if -1.1000000000000001e-6 < b < 6.7999999999999997e-9

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in b around 0 99.3%

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

      1. associate-/l* 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6} \lor \neg \left(b \leq 6.8 \cdot 10^{-9}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% 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 80.1%

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

   1. +-commutative 80.1%

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

3. Simplified 80.1%

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

5. Taylor expanded in a around 0 63.7%

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

   1. associate-/l* 63.6%

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

   2. *-commutative 63.6%

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

   3. quot-tan 63.7%

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

7. Applied egg-rr 63.7%

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

8. Final simplification 63.7%

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

Alternative 9: 39.0% 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 80.1%

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

   1. +-commutative 80.1%

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

3. Simplified 80.1%

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

   1. add-log-exp 40.9%

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

6. Applied egg-rr 40.9%

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

7. Taylor expanded in b around 0 18.6%

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

8. Taylor expanded in a around 0 41.6%

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

Alternative 10: 34.9% 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 80.1%

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

   1. +-commutative 80.1%

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

3. Simplified 80.1%

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

5. Taylor expanded in b around 0 53.7%

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

   1. associate-/l* 53.7%

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

7. Simplified 53.7%

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

8. Taylor expanded in a around 0 37.0%

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

   1. *-commutative 37.0%

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

10. Simplified 37.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))