rsin B (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 11.2s

Alternatives: 13

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: 76.9% 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

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

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 75.5%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. 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. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. 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 2: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ \mathbf{if}\;t\_0 \leq -0.08 \lor \neg \left(t\_0 \leq 0.01\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))))
   (if (or (<= t_0 -0.08) (not (<= t_0 0.01)))
     (* (/ r (cos b)) (sin b))
     (/ (* (sin b) r) (cos a)))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double tmp;
	if ((t_0 <= -0.08) || !(t_0 <= 0.01)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = (sin(b) * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    if ((t_0 <= (-0.08d0)) .or. (.not. (t_0 <= 0.01d0))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = (sin(b) * r) / cos(a)
    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 <= -0.08) || !(t_0 <= 0.01)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	tmp = 0
	if (t_0 <= -0.08) or not (t_0 <= 0.01):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (math.sin(b) * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	tmp = 0.0
	if ((t_0 <= -0.08) || !(t_0 <= 0.01))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	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 <= -0.08) || ~((t_0 <= 0.01)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (sin(b) * r) / cos(a);
	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, -0.08], N[Not[LessEqual[t$95$0, 0.01]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0800000000000000017 or 0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 54.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 53.2

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

   5. Applied rewrites 53.2%

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

   if -0.0800000000000000017 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.0100000000000000002

   1. Initial program 97.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. 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 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 97.5

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

   7. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.6

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

   9. Applied rewrites 97.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.08 \lor \neg \left(\frac{\sin b}{\cos \left(a + b\right)} \leq 0.01\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ \mathbf{if}\;t\_0 \leq -0.08 \lor \neg \left(t\_0 \leq 0.01\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))))
   (if (or (<= t_0 -0.08) (not (<= t_0 0.01)))
     (* (/ r (cos b)) (sin b))
     (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double tmp;
	if ((t_0 <= -0.08) || !(t_0 <= 0.01)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = r * (sin(b) / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    if ((t_0 <= (-0.08d0)) .or. (.not. (t_0 <= 0.01d0))) then
        tmp = (r / cos(b)) * sin(b)
    else
        tmp = r * (sin(b) / cos(a))
    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 <= -0.08) || !(t_0 <= 0.01)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	tmp = 0
	if (t_0 <= -0.08) or not (t_0 <= 0.01):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	tmp = 0.0
	if ((t_0 <= -0.08) || !(t_0 <= 0.01))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	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 <= -0.08) || ~((t_0 <= 0.01)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = r * (sin(b) / cos(a));
	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, -0.08], N[Not[LessEqual[t$95$0, 0.01]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0800000000000000017 or 0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 54.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 53.2

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

   5. Applied rewrites 53.2%

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

   if -0.0800000000000000017 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.0100000000000000002

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.08 \lor \neg \left(\frac{\sin b}{\cos \left(a + b\right)} \leq 0.01\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.34 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.34) (not (<= b 1.08e-8)))
   (* (/ r (cos b)) (sin b))
   (* r (/ (fma (* b b) (* b -0.16666666666666666) b) (cos (+ a b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.34) || !(b <= 1.08e-8)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = r * (fma((b * b), (b * -0.16666666666666666), b) / cos((a + b)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.34) || !(b <= 1.08e-8))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(r * Float64(fma(Float64(b * b), Float64(b * -0.16666666666666666), b) / cos(Float64(a + b))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.34], N[Not[LessEqual[b, 1.08e-8]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.340000000000000024 or 1.0800000000000001e-8 < b

   1. Initial program 54.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 53.6

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

   5. Applied rewrites 53.6%

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

   if -0.340000000000000024 < b < 1.0800000000000001e-8

   1. Initial program 97.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right)}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.34 \lor \neg \left(b \leq 1.08 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% 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 75.5%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 75.5

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

4. Applied rewrites 75.5%

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

Alternative 6: 76.9% 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 75.5%

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

Alternative 7: 53.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 1.08e-8) (* r (/ b (cos a))) (* (/ r 1.0) (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 1.08e-8) {
		tmp = r * (b / cos(a));
	} else {
		tmp = (r / 1.0) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.08d-8) then
        tmp = r * (b / cos(a))
    else
        tmp = (r / 1.0d0) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 1.08e-8) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = (r / 1.0) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 1.08e-8:
		tmp = r * (b / math.cos(a))
	else:
		tmp = (r / 1.0) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 1.08e-8)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(Float64(r / 1.0) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 1.08e-8)
		tmp = r * (b / cos(a));
	else
		tmp = (r / 1.0) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 1.08e-8], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0800000000000001e-8

   1. Initial program 83.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 65.8

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

   5. Applied rewrites 65.8%

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

   if 1.0800000000000001e-8 < b

   1. Initial program 53.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. 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 53.4

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 13.5

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

   7. Applied rewrites 13.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.9%

       \[\leadsto \frac{r}{1} \cdot \sin b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.3% 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 75.5%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 49.8

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

5. Applied rewrites 49.8%

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

Alternative 9: 51.3% 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 75.5%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 49.8

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

7. Applied rewrites 49.8%

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

Alternative 10: 34.9% 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 75.5%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 34.1%

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

Alternative 11: 33.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 49.6

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

5. Applied rewrites 49.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.5%

   \[\leadsto r \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{a}, b\right) \]
9. Add Preprocessing

Alternative 12: 32.7% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 49.6

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

5. Applied rewrites 49.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.5%

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

10. Applied rewrites 32.4%

    \[\leadsto r \cdot \mathsf{fma}\left(a \cdot b, b, b\right) \]
11. Add Preprocessing

Alternative 13: 14.4% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 49.6

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

5. Applied rewrites 49.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.5%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 17.0%

    \[\leadsto r \cdot \left(\left(b \cdot b\right) \cdot a\right) \]
12. Step-by-step derivation

13. Applied rewrites 17.0%

    \[\leadsto r \cdot \left(\left(a \cdot b\right) \cdot b\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))