rsin A (should all be same)

Percentage Accurate: 76.0% → 99.5%

Time: 9.2s

Alternatives: 10

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: 76.0% 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

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

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(Float64(-sin(b)) * 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 75.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / ((cos(b) * cos(a)) - (sin(a) * sin(b)));
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[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-6} \lor \neg \left(b \leq 1.26 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.7e-6) (not (<= b 1.26e-20)))
   (* (/ r (cos b)) (sin b))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.7e-6) || !(b <= 1.26e-20)) {
		tmp = (r / cos(b)) * sin(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 <= (-3.7d-6)) .or. (.not. (b <= 1.26d-20))) then
        tmp = (r / cos(b)) * sin(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 <= -3.7e-6) || !(b <= 1.26e-20)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3.7e-6) or not (b <= 1.26e-20):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -3.7e-6) || !(b <= 1.26e-20))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	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 <= -3.7e-6) || ~((b <= 1.26e-20)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.7e-6], N[Not[LessEqual[b, 1.26e-20]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.7000000000000002e-6 or 1.26e-20 < b

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if -3.7000000000000002e-6 < b < 1.26e-20

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-6} \lor \neg \left(b \leq 1.26 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-20}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3.7e-6)
   (* (/ r (cos b)) (sin b))
   (if (<= b 1.26e-20) (/ (* b r) (cos a)) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.7e-6) {
		tmp = (r / cos(b)) * sin(b);
	} else if (b <= 1.26e-20) {
		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 <= (-3.7d-6)) then
        tmp = (r / cos(b)) * sin(b)
    else if (b <= 1.26d-20) 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 <= -3.7e-6) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else if (b <= 1.26e-20) {
		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 <= -3.7e-6:
		tmp = (r / math.cos(b)) * math.sin(b)
	elif b <= 1.26e-20:
		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 <= -3.7e-6)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	elseif (b <= 1.26e-20)
		tmp = Float64(Float64(b * 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 <= -3.7e-6)
		tmp = (r / cos(b)) * sin(b);
	elseif (b <= 1.26e-20)
		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, -3.7e-6], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e-20], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $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 < -3.7000000000000002e-6

   1. Initial program 59.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if -3.7000000000000002e-6 < b < 1.26e-20

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.9%

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

   if 1.26e-20 < b

   1. Initial program 56.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 57.0

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

   5. Applied rewrites 57.0%

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

Alternative 5: 76.0% 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]

Derivation

1. Initial program 75.4%

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

Alternative 6: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 75.4

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

4. Applied rewrites 75.4%

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

Alternative 7: 54.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.39 \lor \neg \left(b \leq 118\right):\\ \;\;\;\;\left(-r\right) \cdot \left(\sin b \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.39) (not (<= b 118.0)))
   (* (- r) (* (sin b) -1.0))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.39) || !(b <= 118.0)) {
		tmp = -r * (sin(b) * -1.0);
	} 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 <= (-0.39d0)) .or. (.not. (b <= 118.0d0))) then
        tmp = -r * (sin(b) * (-1.0d0))
    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 <= -0.39) || !(b <= 118.0)) {
		tmp = -r * (Math.sin(b) * -1.0);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.39) or not (b <= 118.0):
		tmp = -r * (math.sin(b) * -1.0)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.39) || !(b <= 118.0))
		tmp = Float64(Float64(-r) * Float64(sin(b) * -1.0));
	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 <= -0.39) || ~((b <= 118.0)))
		tmp = -r * (sin(b) * -1.0);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.39], N[Not[LessEqual[b, 118.0]], $MachinePrecision]], N[((-r) * N[(N[Sin[b], $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 118 < b

   1. Initial program 56.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto \left(-r\right) \cdot \left(\sin b \cdot -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 12.7%

       \[\leadsto \left(-r\right) \cdot \left(\sin b \cdot -1\right) \]

   if -0.39000000000000001 < b < 118

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.39 \lor \neg \left(b \leq 118\right):\\ \;\;\;\;\left(-r\right) \cdot \left(\sin b \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.39 \lor \neg \left(b \leq 118\right):\\ \;\;\;\;\left(-r\right) \cdot \left(\sin b \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.39) (not (<= b 118.0)))
   (* (- r) (* (sin b) -1.0))
   (* (/ b (cos a)) r)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.39) || !(b <= 118.0)) {
		tmp = -r * (sin(b) * -1.0);
	} else {
		tmp = (b / cos(a)) * r;
	}
	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 <= (-0.39d0)) .or. (.not. (b <= 118.0d0))) then
        tmp = -r * (sin(b) * (-1.0d0))
    else
        tmp = (b / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.39) || !(b <= 118.0)) {
		tmp = -r * (Math.sin(b) * -1.0);
	} else {
		tmp = (b / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.39) or not (b <= 118.0):
		tmp = -r * (math.sin(b) * -1.0)
	else:
		tmp = (b / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.39) || !(b <= 118.0))
		tmp = Float64(Float64(-r) * Float64(sin(b) * -1.0));
	else
		tmp = Float64(Float64(b / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.39) || ~((b <= 118.0)))
		tmp = -r * (sin(b) * -1.0);
	else
		tmp = (b / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.39], N[Not[LessEqual[b, 118.0]], $MachinePrecision]], N[((-r) * N[(N[Sin[b], $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.39000000000000001 or 118 < b

   1. Initial program 56.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 11.6

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

   7. Applied rewrites 11.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto \left(-r\right) \cdot \left(\sin b \cdot -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 12.7%

       \[\leadsto \left(-r\right) \cdot \left(\sin b \cdot -1\right) \]

   if -0.39000000000000001 < b < 118

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.39 \lor \neg \left(b \leq 118\right):\\ \;\;\;\;\left(-r\right) \cdot \left(\sin b \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(-r\right) \cdot \left(\sin b \cdot -1\right) \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (- r) (* (sin b) -1.0)))

C

double code(double r, double a, double b) {
	return -r * (sin(b) * -1.0);
}

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) * (-1.0d0))
end function

Java

public static double code(double r, double a, double b) {
	return -r * (Math.sin(b) * -1.0);
}

Python

def code(r, a, b):
	return -r * (math.sin(b) * -1.0)

Julia

function code(r, a, b)
	return Float64(Float64(-r) * Float64(sin(b) * -1.0))
end

MATLAB

function tmp = code(r, a, b)
	tmp = -r * (sin(b) * -1.0);
end

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 75.3

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

4. Applied rewrites 75.3%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 49.8

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

7. Applied rewrites 49.8%

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

8. Taylor expanded in a around 0

   \[\leadsto \left(-r\right) \cdot \left(\sin b \cdot -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 34.9%

    \[\leadsto \left(-r\right) \cdot \left(\sin b \cdot -1\right) \]
11. Add Preprocessing

Alternative 10: 34.5% accurate, 36.7× 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 75.4%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 45.4

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 29.9%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))