rsin A (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 20.8s

Alternatives: 9

Speedup: 0.4×

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: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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) / ((cos(b) * cos(a)) - (sin(b) * sin(a)))) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 77.9%

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

   3. +-commutative 77.9%

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

3. Simplified 77.9%

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

   1. associate-/r/ 78.0%

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

5. Applied egg-rr 78.0%

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

   1. cos-sum 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00066 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00066) (not (<= a 1.15e-14)))
   (* (sin b) (/ r (cos a)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00066) || !(a <= 1.15e-14)) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = sin(b) * (r / 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 <= (-0.00066d0)) .or. (.not. (a <= 1.15d-14))) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00066) || !(a <= 1.15e-14)) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00066) or not (a <= 1.15e-14):
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00066) || !(a <= 1.15e-14))
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00066) || ~((a <= 1.15e-14)))
		tmp = sin(b) * (r / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00066], N[Not[LessEqual[a, 1.15e-14]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6e-4 or 1.14999999999999999e-14 < a

   1. Initial program 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/l* 58.8%

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

      3. +-commutative 58.8%

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

   3. Simplified 58.8%

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

      1. add-exp-log 25.9%

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

   5. Applied egg-rr 25.9%

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

      1. div-inv 25.9%

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

      2. add-exp-log 58.6%

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

   7. Applied egg-rr 58.6%

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

   8. Taylor expanded in b around 0 58.7%

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

   if -6.6e-4 < a < 1.14999999999999999e-14

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in a around 0 98.6%

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

      1. associate-/l* 98.5%

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

      2. associate-/r/ 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00066 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00136 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00136) (not (<= a 1.15e-14)))
   (/ (sin b) (/ (cos a) r))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00136) || !(a <= 1.15e-14)) {
		tmp = sin(b) / (cos(a) / r);
	} else {
		tmp = sin(b) * (r / 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 <= (-0.00136d0)) .or. (.not. (a <= 1.15d-14))) then
        tmp = sin(b) / (cos(a) / r)
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00136) || !(a <= 1.15e-14)) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00136) or not (a <= 1.15e-14):
		tmp = math.sin(b) / (math.cos(a) / r)
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00136) || !(a <= 1.15e-14))
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00136) || ~((a <= 1.15e-14)))
		tmp = sin(b) / (cos(a) / r);
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00136], N[Not[LessEqual[a, 1.15e-14]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00136 or 1.14999999999999999e-14 < a

   1. Initial program 58.7%

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

      1. *-commutative 58.7%

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

      2. associate-/l* 58.8%

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

      3. +-commutative 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in b around 0 58.7%

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

   if -0.00136 < a < 1.14999999999999999e-14

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in a around 0 98.6%

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

      1. associate-/l* 98.5%

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

      2. associate-/r/ 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00136 \lor \neg \left(a \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 4: 76.4% 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 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 77.9%

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

   3. +-commutative 77.9%

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

3. Simplified 77.9%

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

   1. clear-num 77.3%

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

   2. associate-/r/ 77.8%

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

   3. clear-num 78.0%

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

5. Applied egg-rr 78.0%

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

6. Final simplification 78.0%

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

Alternative 5: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 77.9%

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

   3. +-commutative 77.9%

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

3. Simplified 77.9%

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

   1. add-exp-log 35.2%

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

5. Applied egg-rr 35.2%

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

   1. div-inv 35.2%

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

   2. add-exp-log 77.8%

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

7. Applied egg-rr 77.8%

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

8. Taylor expanded in b around 0 55.8%

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

9. Final simplification 55.8%

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

Alternative 6: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 9200000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.68) (not (<= b 9200000.0)))
   (* (sin b) r)
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.68) || !(b <= 9200000.0)) {
		tmp = sin(b) * r;
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.68d0)) .or. (.not. (b <= 9200000.0d0))) then
        tmp = sin(b) * r
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.68) || !(b <= 9200000.0)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.68) or not (b <= 9200000.0):
		tmp = math.sin(b) * r
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.68) || !(b <= 9200000.0))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.68) || ~((b <= 9200000.0)))
		tmp = sin(b) * r;
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.68], N[Not[LessEqual[b, 9200000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.680000000000000049 or 9.2e6 < b

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-/l* 55.7%

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

      3. +-commutative 55.7%

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

   3. Simplified 55.7%

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

      1. add-exp-log 26.8%

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

   5. Applied egg-rr 26.8%

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

   6. Taylor expanded in b around 0 5.8%

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

      1. +-commutative 5.8%

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

      2. mul-1-neg 5.8%

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

      3. unsub-neg 5.8%

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

      4. associate-/l* 6.1%

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

   8. Simplified 6.1%

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

   9. Taylor expanded in a around 0 11.1%

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

       1. *-commutative 11.1%

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

   11. Simplified 11.1%

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

   if -0.680000000000000049 < b < 9.2e6

   1. Initial program 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/l* 98.8%

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

      3. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 97.6%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 9200000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 7: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 9200000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.68) (not (<= b 9200000.0)))
   (* (sin b) r)
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.68) || !(b <= 9200000.0)) {
		tmp = sin(b) * r;
	} else {
		tmp = (b * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.68d0)) .or. (.not. (b <= 9200000.0d0))) then
        tmp = sin(b) * r
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.68) || !(b <= 9200000.0)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.68) or not (b <= 9200000.0):
		tmp = math.sin(b) * r
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.68) || !(b <= 9200000.0))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.68) || ~((b <= 9200000.0)))
		tmp = sin(b) * r;
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.68], N[Not[LessEqual[b, 9200000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.680000000000000049 or 9.2e6 < b

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-/l* 55.7%

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

      3. +-commutative 55.7%

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

   3. Simplified 55.7%

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

      1. add-exp-log 26.8%

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

   5. Applied egg-rr 26.8%

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

   6. Taylor expanded in b around 0 5.8%

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

      1. +-commutative 5.8%

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

      2. mul-1-neg 5.8%

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

      3. unsub-neg 5.8%

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

      4. associate-/l* 6.1%

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

   8. Simplified 6.1%

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

   9. Taylor expanded in a around 0 11.1%

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

       1. *-commutative 11.1%

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

   11. Simplified 11.1%

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

   if -0.680000000000000049 < b < 9.2e6

   1. Initial program 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/l* 98.8%

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

      3. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 97.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 9200000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 8: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 77.9%

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

   3. +-commutative 77.9%

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

3. Simplified 77.9%

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

   1. add-exp-log 35.2%

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

5. Applied egg-rr 35.2%

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

6. Taylor expanded in b around 0 53.2%

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

   1. +-commutative 53.2%

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

   2. mul-1-neg 53.2%

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

   3. unsub-neg 53.2%

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

   4. associate-/l* 53.4%

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

8. Simplified 53.4%

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

9. Taylor expanded in a around 0 37.4%

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

    1. *-commutative 37.4%

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

11. Simplified 37.4%

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

12. Final simplification 37.4%

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

Alternative 9: 34.3% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 77.9%

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

   3. +-commutative 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in b around 0 52.0%

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

   1. associate-/l* 51.9%

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

   2. associate-/r/ 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in a around 0 33.8%

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

   1. *-commutative 33.8%

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

9. Simplified 33.8%

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

10. Final simplification 33.8%

    \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023279 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))