rsin B (should all be same)

Percentage Accurate: 76.0% → 99.5%

Time: 9.8s

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: 76.0% 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 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.4

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

4. Applied rewrites 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
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 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ \mathbf{if}\;t\_1 \leq -0.005 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b))) (t_1 (/ (sin b) t_0)))
   (if (or (<= t_1 -0.005) (not (<= t_1 5e-22)))
     (* (/ r (cos b)) (sin b))
     (/ (* r b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = cos((a + b));
	double t_1 = sin(b) / t_0;
	double tmp;
	if ((t_1 <= -0.005) || !(t_1 <= 5e-22)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (r * b) / t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = cos((a + b))
    t_1 = sin(b) / t_0
    if ((t_1 <= (-0.005d0)) .or. (.not. (t_1 <= 5d-22))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (r * b) / t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.cos((a + b));
	double t_1 = Math.sin(b) / t_0;
	double tmp;
	if ((t_1 <= -0.005) || !(t_1 <= 5e-22)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (r * b) / t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.cos((a + b))
	t_1 = math.sin(b) / t_0
	tmp = 0
	if (t_1 <= -0.005) or not (t_1 <= 5e-22):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (r * b) / t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = cos(Float64(a + b))
	t_1 = Float64(sin(b) / t_0)
	tmp = 0.0
	if ((t_1 <= -0.005) || !(t_1 <= 5e-22))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(r * b) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = cos((a + b));
	t_1 = sin(b) / t_0;
	tmp = 0.0;
	if ((t_1 <= -0.005) || ~((t_1 <= 5e-22)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (r * b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.005], N[Not[LessEqual[t$95$1, 5e-22]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0050000000000000001 or 4.99999999999999954e-22 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 57.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 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 -0.0050000000000000001 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.99999999999999954e-22

   1. Initial program 99.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.005 \lor \neg \left(\frac{\sin b}{\cos \left(a + b\right)} \leq 5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ \mathbf{if}\;t\_1 \leq -0.005:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{r \cdot b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b))) (t_1 (/ (sin b) t_0)))
   (if (<= t_1 -0.005)
     (/ (* (sin b) r) (cos b))
     (if (<= t_1 5e-22) (/ (* r b) t_0) (* (/ r (cos b)) (sin b))))))

C

double code(double r, double a, double b) {
	double t_0 = cos((a + b));
	double t_1 = sin(b) / t_0;
	double tmp;
	if (t_1 <= -0.005) {
		tmp = (sin(b) * r) / cos(b);
	} else if (t_1 <= 5e-22) {
		tmp = (r * b) / t_0;
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((a + b))
    t_1 = sin(b) / t_0
    if (t_1 <= (-0.005d0)) then
        tmp = (sin(b) * r) / cos(b)
    else if (t_1 <= 5d-22) then
        tmp = (r * b) / t_0
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.cos((a + b));
	double t_1 = Math.sin(b) / t_0;
	double tmp;
	if (t_1 <= -0.005) {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	} else if (t_1 <= 5e-22) {
		tmp = (r * b) / t_0;
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.cos((a + b))
	t_1 = math.sin(b) / t_0
	tmp = 0
	if t_1 <= -0.005:
		tmp = (math.sin(b) * r) / math.cos(b)
	elif t_1 <= 5e-22:
		tmp = (r * b) / t_0
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = cos(Float64(a + b))
	t_1 = Float64(sin(b) / t_0)
	tmp = 0.0
	if (t_1 <= -0.005)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	elseif (t_1 <= 5e-22)
		tmp = Float64(Float64(r * b) / t_0);
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = cos((a + b));
	t_1 = sin(b) / t_0;
	tmp = 0.0;
	if (t_1 <= -0.005)
		tmp = (sin(b) * r) / cos(b);
	elseif (t_1 <= 5e-22)
		tmp = (r * b) / t_0;
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.005], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-22], N[(N[(r * b), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0050000000000000001

   1. Initial program 57.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.3

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 57.6

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

   7. Applied rewrites 57.6%

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

   if -0.0050000000000000001 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.99999999999999954e-22

   1. Initial program 99.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 4.99999999999999954e-22 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 58.1%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

Alternative 4: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ t_1 := \frac{\sin b}{t\_0}\\ \mathbf{if}\;t\_1 \leq -0.005:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{r \cdot b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b))) (t_1 (/ (sin b) t_0)))
   (if (<= t_1 -0.005)
     (* r (/ (sin b) (cos b)))
     (if (<= t_1 5e-22) (/ (* r b) t_0) (* (/ r (cos b)) (sin b))))))

C

double code(double r, double a, double b) {
	double t_0 = cos((a + b));
	double t_1 = sin(b) / t_0;
	double tmp;
	if (t_1 <= -0.005) {
		tmp = r * (sin(b) / cos(b));
	} else if (t_1 <= 5e-22) {
		tmp = (r * b) / t_0;
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((a + b))
    t_1 = sin(b) / t_0
    if (t_1 <= (-0.005d0)) then
        tmp = r * (sin(b) / cos(b))
    else if (t_1 <= 5d-22) then
        tmp = (r * b) / t_0
    else
        tmp = (r / cos(b)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.cos((a + b));
	double t_1 = Math.sin(b) / t_0;
	double tmp;
	if (t_1 <= -0.005) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (t_1 <= 5e-22) {
		tmp = (r * b) / t_0;
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.cos((a + b))
	t_1 = math.sin(b) / t_0
	tmp = 0
	if t_1 <= -0.005:
		tmp = r * (math.sin(b) / math.cos(b))
	elif t_1 <= 5e-22:
		tmp = (r * b) / t_0
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = cos(Float64(a + b))
	t_1 = Float64(sin(b) / t_0)
	tmp = 0.0
	if (t_1 <= -0.005)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (t_1 <= 5e-22)
		tmp = Float64(Float64(r * b) / t_0);
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = cos((a + b));
	t_1 = sin(b) / t_0;
	tmp = 0.0;
	if (t_1 <= -0.005)
		tmp = r * (sin(b) / cos(b));
	elseif (t_1 <= 5e-22)
		tmp = (r * b) / t_0;
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.005], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-22], N[(N[(r * b), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0050000000000000001

   1. Initial program 57.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if -0.0050000000000000001 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.99999999999999954e-22

   1. Initial program 99.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 4.99999999999999954e-22 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 58.1%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.7

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

   5. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{r}{\cos b} \cdot \sin b} \]
3. Recombined 3 regimes into one program.
4. 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 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.4

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

4. Applied rewrites 99.4%

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

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. associate-*r* N/A

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

   6. mul-1-neg 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. lower-fma.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-cos.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 6: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 75.4

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

4. Applied rewrites 75.4%

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

Alternative 7: 76.0% 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 75.4%

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

Alternative 8: 51.3% 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 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 75.2

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

4. Applied rewrites 75.2%

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

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 45.9

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

7. Applied rewrites 45.9%

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

Alternative 9: 50.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 75.4

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

4. Applied rewrites 75.4%

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

5. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 45.6

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

7. Applied rewrites 45.6%

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

Alternative 10: 50.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 45.5

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

5. Applied rewrites 45.5%

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

Alternative 11: 50.9% 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 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.4

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

4. Applied rewrites 99.4%

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

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 45.4

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

7. Applied rewrites 45.4%

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

Alternative 12: 34.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{b}{1} \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return r * (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 * (b / 1.0d0)
end function

Java

public static double code(double r, double a, double b) {
	return r * (b / 1.0);
}

Python

def code(r, a, b):
	return r * (b / 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 45.5

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

5. Applied rewrites 45.5%

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

6. Taylor expanded in a around 0

   \[\leadsto r \cdot \frac{b}{1} \]
7. Step-by-step derivation

8. Applied rewrites 29.9%

   \[\leadsto r \cdot \frac{b}{1} \]
9. Add Preprocessing

Reproduce

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