rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 16.6s

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: 76.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 78.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 78.9

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

4. Applied rewrites 78.9%

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

   1. cos-sum N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 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. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. cos-sum N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -9 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -9e-6) t_0 (if (<= a 1.46e-5) (* r (/ (sin b) (cos b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(a));
	double tmp;
	if (a <= -9e-6) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = 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) :: tmp
    t_0 = sin(b) * (r / cos(a))
    if (a <= (-9d-6)) then
        tmp = t_0
    else if (a <= 1.46d-5) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * (r / Math.cos(a));
	double tmp;
	if (a <= -9e-6) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(a))
	tmp = 0
	if a <= -9e-6:
		tmp = t_0
	elif a <= 1.46e-5:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(a)))
	tmp = 0.0
	if (a <= -9e-6)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(a));
	tmp = 0.0;
	if (a <= -9e-6)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9e-6], t$95$0, If[LessEqual[a, 1.46e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.00000000000000023e-6 or 1.46000000000000008e-5 < a

   1. Initial program 58.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 58.2

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -9.00000000000000023e-6 < a < 1.46000000000000008e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -9 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -9e-6) t_0 (if (<= a 1.46e-5) (* r (/ (sin b) (cos b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -9e-6) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = 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) :: tmp
    t_0 = r * (sin(b) / cos(a))
    if (a <= (-9d-6)) then
        tmp = t_0
    else if (a <= 1.46d-5) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / Math.cos(a));
	double tmp;
	if (a <= -9e-6) {
		tmp = t_0;
	} else if (a <= 1.46e-5) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -9e-6:
		tmp = t_0
	elif a <= 1.46e-5:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -9e-6)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -9e-6)
		tmp = t_0;
	elseif (a <= 1.46e-5)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9e-6], t$95$0, If[LessEqual[a, 1.46e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.00000000000000023e-6 or 1.46000000000000008e-5 < a

   1. Initial program 58.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   if -9.00000000000000023e-6 < a < 1.46000000000000008e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 78.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 78.9

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

4. Applied rewrites 78.9%

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

5. Final simplification 78.9%

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

Alternative 7: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Final simplification 78.9%

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

Alternative 8: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 56.8

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

5. Applied rewrites 56.8%

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

Alternative 9: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 2.7)
   (*
    (/ r (cos (+ b a)))
    (fma
     (fma (* b b) 0.008333333333333333 -0.16666666666666666)
     (* b (* b b))
     b))
   (/ 1.0 (/ 1.0 (* r (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 2.7) {
		tmp = (r / cos((b + a))) * fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * (b * b)), b);
	} else {
		tmp = 1.0 / (1.0 / (r * sin(b)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 2.7)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * Float64(b * b)), b));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(r * sin(b))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 2.7], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 88.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 88.1

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot b + 1 \cdot b\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\color{blue}{\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2}\right)} \cdot b + 1 \cdot b\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \left({b}^{2} \cdot b\right)} + 1 \cdot b\right) \]

      5. pow-plus N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{b}^{\left(2 + 1\right)}} + 1 \cdot b\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{\color{blue}{3}} + 1 \cdot b\right) \]

      7. cube-unmult N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} + 1 \cdot b\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \left(b \cdot \color{blue}{{b}^{2}}\right) + 1 \cdot b\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right) + \color{blue}{b}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, b \cdot {b}^{2}, b\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, b \cdot {b}^{2}, b\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\color{blue}{{b}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), b \cdot {b}^{2}, b\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, b \cdot {b}^{2}, b\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, b \cdot {b}^{2}, b\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120}, \frac{-1}{6}\right), b \cdot {b}^{2}, b\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120}, \frac{-1}{6}\right), b \cdot {b}^{2}, b\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{b \cdot {b}^{2}}, b\right) \]

      18. unpow2 N/A

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

      19. lower-*.f64 70.0

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

   7. Applied rewrites 70.0%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)} \]

   if 2.7000000000000002 < b

   1. Initial program 51.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.3

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 53.0

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

   9. Applied rewrites 53.0%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 11.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 2.7)
   (*
    r
    (/
     (fma
      (fma (* b b) 0.008333333333333333 -0.16666666666666666)
      (* b (* b b))
      b)
     (cos (+ b a))))
   (/ 1.0 (/ 1.0 (* r (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 2.7) {
		tmp = r * (fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / (r * sin(b)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 2.7)
		tmp = Float64(r * Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(r * sin(b))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 2.7], N[(r * N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 88.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right)}}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) + b \cdot 1}}{\cos \left(a + b\right)} \]

      3. associate-*r* N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left(b \cdot {b}^{2}\right) \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto r \cdot \frac{\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right) + \color{blue}{b}}{\cos \left(a + b\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, b \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

      7. sub-neg N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, b \cdot {b}^{2}, b\right)}{\cos \left(a + b\right)} \]

      8. *-commutative N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), b \cdot {b}^{2}, b\right)}{\cos \left(a + b\right)} \]

      9. metadata-eval N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, b \cdot {b}^{2}, b\right)}{\cos \left(a + b\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, b \cdot {b}^{2}, b\right)}{\cos \left(a + b\right)} \]

      11. unpow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120}, \frac{-1}{6}\right), b \cdot {b}^{2}, b\right)}{\cos \left(a + b\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120}, \frac{-1}{6}\right), b \cdot {b}^{2}, b\right)}{\cos \left(a + b\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{b \cdot {b}^{2}}, b\right)}{\cos \left(a + b\right)} \]

      14. unpow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), b \cdot \color{blue}{\left(b \cdot b\right)}, b\right)}{\cos \left(a + b\right)} \]

      15. lower-*.f64 69.9

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \color{blue}{\left(b \cdot b\right)}, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 69.9%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(a + b\right)} \]

   if 2.7000000000000002 < b

   1. Initial program 51.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.3

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 53.0

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

   9. Applied rewrites 53.0%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 11.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 460000000000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 460000000000.0) (* b (/ r (cos a))) (/ 1.0 (/ 1.0 (* r (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 460000000000.0) {
		tmp = b * (r / cos(a));
	} else {
		tmp = 1.0 / (1.0 / (r * 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 <= 460000000000.0d0) then
        tmp = b * (r / cos(a))
    else
        tmp = 1.0d0 / (1.0d0 / (r * sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 460000000000.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = 1.0 / (1.0 / (r * Math.sin(b)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 460000000000.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = 1.0 / (1.0 / (r * math.sin(b)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 460000000000.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(r * sin(b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 460000000000.0)
		tmp = b * (r / cos(a));
	else
		tmp = 1.0 / (1.0 / (r * sin(b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 460000000000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6e11

   1. Initial program 87.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 4.6e11 < b

   1. Initial program 50.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-cos.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 12.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 460000000000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 53.4

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

5. Applied rewrites 53.4%

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

Alternative 13: 35.4% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 35.6

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

8. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))