rsin B (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 12.5s

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: 76.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 78.8

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

4. Applied rewrites 78.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. cancel-sign-sub-inv N/A

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

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

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

   12. distribute-rgt-neg-out N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-neg.f64 N/A

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

   19. lower-fma.f64 99.6

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))))
   (if (<= t_0 -0.02)
     (* (/ r (cos b)) (sin b))
     (if (<= t_0 0.0005)
       (* (/ (sin b) (cos a)) r)
       (/ (* (sin b) r) (cos b))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (r / cos(b)) * sin(b);
	} else if (t_0 <= 0.0005) {
		tmp = (sin(b) / cos(a)) * r;
	} else {
		tmp = (sin(b) * r) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    if (t_0 <= (-0.02d0)) then
        tmp = (r / cos(b)) * sin(b)
    else if (t_0 <= 0.0005d0) then
        tmp = (sin(b) / cos(a)) * r
    else
        tmp = (sin(b) * r) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) / Math.cos((a + b));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else if (t_0 <= 0.0005) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	tmp = 0
	if t_0 <= -0.02:
		tmp = (r / math.cos(b)) * math.sin(b)
	elif t_0 <= 0.0005:
		tmp = (math.sin(b) / math.cos(a)) * r
	else:
		tmp = (math.sin(b) * r) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	elseif (t_0 <= 0.0005)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	else
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((a + b));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = (r / cos(b)) * sin(b);
	elseif (t_0 <= 0.0005)
		tmp = (sin(b) / cos(a)) * r;
	else
		tmp = (sin(b) * r) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0005], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0200000000000000004

   1. Initial program 59.7%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 5.0000000000000001e-4

   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 5.0000000000000001e-4 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 58.1%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.1

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

   7. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 58.2

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

   9. Applied rewrites 58.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.02:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 0.0005:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ t_1 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))) (t_1 (* (/ r (cos b)) (sin b))))
   (if (<= t_0 -0.02) t_1 (if (<= t_0 0.0005) (* (/ (sin b) (cos a)) r) t_1))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double t_1 = (r / cos(b)) * sin(b);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.0005) {
		tmp = (sin(b) / cos(a)) * r;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(b) / cos((a + b))
    t_1 = (r / cos(b)) * sin(b)
    if (t_0 <= (-0.02d0)) then
        tmp = t_1
    else if (t_0 <= 0.0005d0) then
        tmp = (sin(b) / cos(a)) * r
    else
        tmp = t_1
    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 t_1 = (r / Math.cos(b)) * Math.sin(b);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.0005) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	t_1 = (r / math.cos(b)) * math.sin(b)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_1
	elif t_0 <= 0.0005:
		tmp = (math.sin(b) / math.cos(a)) * r
	else:
		tmp = t_1
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	t_1 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.0005)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((a + b));
	t_1 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.0005)
		tmp = (sin(b) / cos(a)) * r;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 0.0005], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0200000000000000004 or 5.0000000000000001e-4 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 59.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 59.4

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

   5. Applied rewrites 59.4%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 5.0000000000000001e-4

   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 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.02:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(a + b\right)} \leq 0.0005:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / fma(sin(a), Float64(-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 78.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in a around inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-cos.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. mul-1-neg N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-sin.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 7: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -520:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.00165:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -520.0) t_0 (if (<= b 0.00165) (* (/ b (cos a)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -520.0) {
		tmp = t_0;
	} else if (b <= 0.00165) {
		tmp = (b / cos(a)) * r;
	} 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 / cos(b)) * sin(b)
    if (b <= (-520.0d0)) then
        tmp = t_0
    else if (b <= 0.00165d0) then
        tmp = (b / cos(a)) * r
    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.cos(b)) * Math.sin(b);
	double tmp;
	if (b <= -520.0) {
		tmp = t_0;
	} else if (b <= 0.00165) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(b)) * math.sin(b)
	tmp = 0
	if b <= -520.0:
		tmp = t_0
	elif b <= 0.00165:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -520.0)
		tmp = t_0;
	elseif (b <= 0.00165)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (b <= -520.0)
		tmp = t_0;
	elseif (b <= 0.00165)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -520.0], t$95$0, If[LessEqual[b, 0.00165], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -520 or 0.00165 < b

   1. Initial program 59.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 59.4

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

   5. Applied rewrites 59.4%

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

   if -520 < b < 0.00165

   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

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -520:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{elif}\;b \leq 0.00165:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% 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 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 78.8

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

4. Applied rewrites 78.8%

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

Alternative 9: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Final simplification 78.8%

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

Alternative 10: 54.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{1} \cdot r\\ \mathbf{if}\;b \leq -0.88:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 62000000:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) 1.0) r)))
   (if (<= b -0.88) t_0 (if (<= b 62000000.0) (* (/ b (cos a)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / 1.0) * r;
	double tmp;
	if (b <= -0.88) {
		tmp = t_0;
	} else if (b <= 62000000.0) {
		tmp = (b / cos(a)) * r;
	} 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) / 1.0d0) * r
    if (b <= (-0.88d0)) then
        tmp = t_0
    else if (b <= 62000000.0d0) then
        tmp = (b / cos(a)) * r
    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) / 1.0) * r;
	double tmp;
	if (b <= -0.88) {
		tmp = t_0;
	} else if (b <= 62000000.0) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / 1.0) * r
	tmp = 0
	if b <= -0.88:
		tmp = t_0
	elif b <= 62000000.0:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / 1.0) * r)
	tmp = 0.0
	if (b <= -0.88)
		tmp = t_0;
	elseif (b <= 62000000.0)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / 1.0) * r;
	tmp = 0.0;
	if (b <= -0.88)
		tmp = t_0;
	elseif (b <= 62000000.0)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.88], t$95$0, If[LessEqual[b, 62000000.0], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.880000000000000004 or 6.2e7 < b

   1. Initial program 59.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 13.8%

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

   if -0.880000000000000004 < b < 6.2e7

   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

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.88:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \mathbf{elif}\;b \leq 62000000:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{1} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 52.0

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

5. Applied rewrites 52.0%

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

6. Final simplification 52.0%

   \[\leadsto \frac{b}{\cos a} \cdot r \]
7. Add Preprocessing

Alternative 12: 50.7% 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 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. inv-pow N/A

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

   14. lower-pow.f64 78.6

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

4. Applied rewrites 78.6%

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

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 52.0

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

7. Applied rewrites 52.0%

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

Alternative 13: 34.1% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b / 1.0d0) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 52.0

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.4%

   \[\leadsto r \cdot \frac{b}{1} \]

9. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))