rsin A (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 14.5s

Alternatives: 14

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: 76.8% 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, 0.3× 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%

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

   1. +-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%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

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%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. 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%

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

   1. +-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%

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

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000195 \lor \neg \left(a \leq 2050000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.000195) (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 <= -0.000195) || !(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 <= (-0.000195d0)) .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 <= -0.000195) || !(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 <= -0.000195) 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 <= -0.000195) || !(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 <= -0.000195) || ~((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, -0.000195], 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 < -1.94999999999999996e-4 or 2.05e12 < a

   1. Initial program 58.9%

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

      1. associate-*r/ 58.9%

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

      2. *-commutative 58.9%

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

      3. +-commutative 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in b around 0 58.5%

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

   if -1.94999999999999996e-4 < a < 2.05e12

   1. Initial program 97.2%

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

      1. associate-*r/ 97.2%

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

      2. *-commutative 97.2%

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

      3. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000195 \lor \neg \left(a \leq 2050000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6: 76.3% accurate, 1.0× speedup

Math

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

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 58.4%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 57.6%

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

   if -2e-3 < a < 2.05e12

   1. Initial program 97.2%

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

      1. associate-*r/ 97.2%

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

      2. *-commutative 97.2%

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

      3. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.1%

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

   if 2.05e12 < a

   1. Initial program 59.4%

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

      1. associate-/l* 59.5%

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

      2. +-commutative 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in b around 0 59.5%

      \[\leadsto \frac{r}{\frac{\color{blue}{\cos a}}{\sin b}} \]
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 7: 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 58.4%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 57.6%

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

   if -1.15e-5 < a < 2.05e12

   1. Initial program 97.2%

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

      1. associate-*r/ 97.2%

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

      2. *-commutative 97.2%

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

      3. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   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 59.4%

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

      1. associate-/l* 59.5%

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

      2. +-commutative 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in b around 0 59.5%

      \[\leadsto \frac{r}{\frac{\color{blue}{\cos a}}{\sin b}} \]
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 8: 76.8% 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 77.0%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-/r/ 76.9%

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

5. Applied egg-rr 76.9%

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

6. Final simplification 76.9%

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

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%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. 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%

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

2. 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%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 55.6%

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

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%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 52.2%

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

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%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. clear-num 76.9%

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

   2. associate-/r/ 76.9%

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

5. Applied egg-rr 76.9%

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

6. Taylor expanded in b around 0 55.5%

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

7. Taylor expanded in a around 0 37.3%

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

8. Final simplification 37.3%

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

Alternative 14: 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%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 52.2%

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

5. Taylor expanded in a around 0 34.1%

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

6. Final simplification 34.1%

   \[\leadsto r \cdot b \]

Reproduce

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