rsin B (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 10.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{--1}{\frac{-1}{\sin b}} \cdot \sin a\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. remove-double-div N/A

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

   2. unpow-1 N/A

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

   3. lift-pow.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow-1 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{--1}{\frac{-1}{\sin b}} \cdot \sin a\right)} \]
8. Add Preprocessing

Alternative 2: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ \mathbf{if}\;t\_0 \leq -0.0002 \lor \neg \left(t\_0 \leq 10^{-7}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left({\cos a}^{-1} \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))))
   (if (or (<= t_0 -0.0002) (not (<= t_0 1e-7)))
     (* (/ r (cos b)) (sin b))
     (* r (* (pow (cos a) -1.0) b)))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double tmp;
	if ((t_0 <= -0.0002) || !(t_0 <= 1e-7)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = r * (pow(cos(a), -1.0) * 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) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    if ((t_0 <= (-0.0002d0)) .or. (.not. (t_0 <= 1d-7))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = r * ((cos(a) ** (-1.0d0)) * b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) / Math.cos((a + b));
	double tmp;
	if ((t_0 <= -0.0002) || !(t_0 <= 1e-7)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = r * (Math.pow(Math.cos(a), -1.0) * b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	tmp = 0
	if (t_0 <= -0.0002) or not (t_0 <= 1e-7):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = r * (math.pow(math.cos(a), -1.0) * b)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	tmp = 0.0
	if ((t_0 <= -0.0002) || !(t_0 <= 1e-7))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(r * Float64((cos(a) ^ -1.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((a + b));
	tmp = 0.0;
	if ((t_0 <= -0.0002) || ~((t_0 <= 1e-7)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = r * ((cos(a) ^ -1.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.0002], N[Not[LessEqual[t$95$0, 1e-7]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Power[N[Cos[a], $MachinePrecision], -1.0], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -2.0000000000000001e-4 or 9.9999999999999995e-8 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 55.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

   if -2.0000000000000001e-4 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 9.9999999999999995e-8

   1. Initial program 99.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto r \cdot \left({\cos a}^{-1} \cdot \color{blue}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.0002 \lor \neg \left(\frac{\sin b}{\cos \left(a + b\right)} \leq 10^{-7}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left({\cos a}^{-1} \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

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(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(Float64(-sin(b)) * sin(a)))))
end

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

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

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

   7. sin-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sin-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-cos.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 6: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 76.8

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

4. Applied rewrites 76.8%

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

Alternative 7: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

Alternative 8: 50.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (/ r (cos (+ a b))) (/ (fma (* b b) 0.16666666666666666 1.0) b)))

C

double code(double r, double a, double b) {
	return (r / cos((a + b))) / (fma((b * b), 0.16666666666666666, 1.0) / b);
}

Julia

function code(r, a, b)
	return Float64(Float64(r / cos(Float64(a + b))) / Float64(fma(Float64(b * b), 0.16666666666666666, 1.0) / b))
end

Wolfram

code[r_, a_, b_] := N[(N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. inv-pow N/A

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

   14. lower-pow.f64 76.6

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

4. Applied rewrites 76.6%

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

5. Taylor expanded in b around 0

   \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot {b}^{2}}{b}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot {b}^{2}}{b}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{\frac{1}{6} \cdot {b}^{2} + 1}}{b}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{{b}^{2} \cdot \frac{1}{6}} + 1}{b}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{6}, 1\right)}}{b}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\frac{\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{6}, 1\right)}{b}} \]

   6. lower-*.f64 51.4

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

7. Applied rewrites 51.4%

   \[\leadsto \frac{\frac{r}{\cos \left(a + b\right)}}{\color{blue}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}} \]
8. Add Preprocessing

Alternative 9: 50.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r}{\cos a}}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (/ r (cos a)) (/ (fma (* b b) 0.16666666666666666 1.0) b)))

C

double code(double r, double a, double b) {
	return (r / cos(a)) / (fma((b * b), 0.16666666666666666, 1.0) / b);
}

Julia

function code(r, a, b)
	return Float64(Float64(r / cos(a)) / Float64(fma(Float64(b * b), 0.16666666666666666, 1.0) / b))
end

Wolfram

code[r_, a_, b_] := N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. inv-pow N/A

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

   14. lower-pow.f64 76.6

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

4. Applied rewrites 76.6%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 54.4

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

7. Applied rewrites 54.4%

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

8. Taylor expanded in b around 0

   \[\leadsto \frac{\frac{r}{\cos a}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot {b}^{2}}{b}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{r}{\cos a}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot {b}^{2}}{b}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{r}{\cos a}}{\frac{\color{blue}{\frac{1}{6} \cdot {b}^{2} + 1}}{b}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{r}{\cos a}}{\frac{\color{blue}{{b}^{2} \cdot \frac{1}{6}} + 1}{b}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{r}{\cos a}}{\frac{\color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{6}, 1\right)}}{b}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{r}{\cos a}}{\frac{\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{6}, 1\right)}{b}} \]

   6. lower-*.f64 51.0

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

10. Applied rewrites 51.0%

    \[\leadsto \frac{\frac{r}{\cos a}}{\color{blue}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}} \]
11. Add Preprocessing

Alternative 10: 50.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(b \cdot b\right), b, b\right)}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (fma (* -0.16666666666666666 (* b b)) b b) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	return r * (fma((-0.16666666666666666 * (b * b)), b, b) / cos((a + b)));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(fma(Float64(-0.16666666666666666 * Float64(b * b)), b, b) / cos(Float64(a + b))))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[(N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b + b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

3. Taylor expanded in b around 0

   \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)}}{\cos \left(a + b\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}}{\cos \left(a + b\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1}}{\cos \left(a + b\right)} \]

   3. *-commutative N/A

      \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left({b}^{2} \cdot \frac{-1}{6}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

   4. associate-*r* N/A

      \[\leadsto r \cdot \frac{\color{blue}{\left(b \cdot {b}^{2}\right) \cdot \frac{-1}{6}} + b \cdot 1}{\cos \left(a + b\right)} \]

   5. *-rgt-identity N/A

      \[\leadsto r \cdot \frac{\left(b \cdot {b}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{b}}{\cos \left(a + b\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(b \cdot {b}^{2}, \frac{-1}{6}, b\right)}}{\cos \left(a + b\right)} \]

   7. unpow2 N/A

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(b \cdot \color{blue}{\left(b \cdot b\right)}, \frac{-1}{6}, b\right)}{\cos \left(a + b\right)} \]

   8. cube-unmult N/A

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, \frac{-1}{6}, b\right)}{\cos \left(a + b\right)} \]

   9. lower-pow.f64 50.8

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, -0.16666666666666666, b\right)}{\cos \left(a + b\right)} \]

5. Applied rewrites 50.8%

   \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, -0.16666666666666666, b\right)}}{\cos \left(a + b\right)} \]
6. Step-by-step derivation

7. Applied rewrites 50.8%

   \[\leadsto r \cdot \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(b \cdot b\right), \color{blue}{b}, b\right)}{\cos \left(a + b\right)} \]
8. Add Preprocessing

Alternative 11: 50.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
6. 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 50.8

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

7. Applied rewrites 50.8%

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

Alternative 12: 35.0% 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 76.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in b around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right) \cdot b} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{r}{\cos a} + \frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}}\right) \cdot b} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{b \cdot \left(r \cdot \sin a\right)}{{\cos a}^{2}} + \frac{r}{\cos a}\right)} \cdot b \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{\left(r \cdot \sin a\right) \cdot b}}{{\cos a}^{2}} + \frac{r}{\cos a}\right) \cdot b \]

   5. associate-/l* N/A

      \[\leadsto \left(\color{blue}{\left(r \cdot \sin a\right) \cdot \frac{b}{{\cos a}^{2}}} + \frac{r}{\cos a}\right) \cdot b \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \sin a, \frac{b}{{\cos a}^{2}}, \frac{r}{\cos a}\right)} \cdot b \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin a \cdot r}, \frac{b}{{\cos a}^{2}}, \frac{r}{\cos a}\right) \cdot b \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin a \cdot r}, \frac{b}{{\cos a}^{2}}, \frac{r}{\cos a}\right) \cdot b \]

   9. lower-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin a} \cdot r, \frac{b}{{\cos a}^{2}}, \frac{r}{\cos a}\right) \cdot b \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin a \cdot r, \color{blue}{\frac{b}{{\cos a}^{2}}}, \frac{r}{\cos a}\right) \cdot b \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin a \cdot r, \frac{b}{\color{blue}{{\cos a}^{2}}}, \frac{r}{\cos a}\right) \cdot b \]

   12. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin a \cdot r, \frac{b}{{\color{blue}{\cos a}}^{2}}, \frac{r}{\cos a}\right) \cdot b \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin a \cdot r, \frac{b}{{\cos a}^{2}}, \color{blue}{\frac{r}{\cos a}}\right) \cdot b \]

   14. lower-cos.f64 50.8

       \[\leadsto \mathsf{fma}\left(\sin a \cdot r, \frac{b}{{\cos a}^{2}}, \frac{r}{\color{blue}{\cos a}}\right) \cdot b \]

7. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin a \cdot r, \frac{b}{{\cos a}^{2}}, \frac{r}{\cos a}\right) \cdot b} \]

8. Taylor expanded in a around 0

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

10. Applied rewrites 36.2%

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

Reproduce

herbie shell --seed 2024323 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))