rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 10.8s

Alternatives: 9

Speedup: 0.4×

Specification

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 79.3

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 79.3

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

4. Applied rewrites 79.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. unsub-neg N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-fma.f64 99.5

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.00011:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -5.3e-5)
   (* (sin b) (/ r (cos a)))
   (if (<= a 0.00011) (* r (/ (sin b) (cos b))) (/ (* (sin b) r) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -5.3e-5) {
		tmp = sin(b) * (r / cos(a));
	} else if (a <= 0.00011) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (sin(b) * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.3d-5)) then
        tmp = sin(b) * (r / cos(a))
    else if (a <= 0.00011d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (sin(b) * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -5.3e-5) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else if (a <= 0.00011) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -5.3e-5:
		tmp = math.sin(b) * (r / math.cos(a))
	elif a <= 0.00011:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (math.sin(b) * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -5.3e-5)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	elseif (a <= 0.00011)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -5.3e-5)
		tmp = sin(b) * (r / cos(a));
	elseif (a <= 0.00011)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (sin(b) * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -5.3e-5], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00011], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.3000000000000001e-5

   1. Initial program 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 63.3

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

   7. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.5

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

   9. Applied rewrites 63.5%

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

   if -5.3000000000000001e-5 < a < 1.10000000000000004e-4

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if 1.10000000000000004e-4 < a

   1. Initial program 49.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 49.6

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

   5. Applied rewrites 49.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.00011:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.00011:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -5.3e-5) t_0 (if (<= a 0.00011) (* 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 <= -5.3e-5) {
		tmp = t_0;
	} else if (a <= 0.00011) {
		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 <= (-5.3d-5)) then
        tmp = t_0
    else if (a <= 0.00011d0) 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 <= -5.3e-5) {
		tmp = t_0;
	} else if (a <= 0.00011) {
		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 <= -5.3e-5:
		tmp = t_0
	elif a <= 0.00011:
		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 <= -5.3e-5)
		tmp = t_0;
	elseif (a <= 0.00011)
		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 <= -5.3e-5)
		tmp = t_0;
	elseif (a <= 0.00011)
		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, -5.3e-5], t$95$0, If[LessEqual[a, 0.00011], 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 < -5.3000000000000001e-5 or 1.10000000000000004e-4 < a

   1. Initial program 55.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.8

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

   9. Applied rewrites 55.8%

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

   if -5.3000000000000001e-5 < a < 1.10000000000000004e-4

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 99.0

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

   7. Applied rewrites 99.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.00011:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.00011:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -5.3e-5) t_0 (if (<= a 0.00011) (* (sin b) (/ r (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 <= -5.3e-5) {
		tmp = t_0;
	} else if (a <= 0.00011) {
		tmp = sin(b) * (r / 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 <= (-5.3d-5)) then
        tmp = t_0
    else if (a <= 0.00011d0) then
        tmp = sin(b) * (r / 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 <= -5.3e-5) {
		tmp = t_0;
	} else if (a <= 0.00011) {
		tmp = Math.sin(b) * (r / 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 <= -5.3e-5:
		tmp = t_0
	elif a <= 0.00011:
		tmp = math.sin(b) * (r / 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 <= -5.3e-5)
		tmp = t_0;
	elseif (a <= 0.00011)
		tmp = Float64(sin(b) * Float64(r / 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 <= -5.3e-5)
		tmp = t_0;
	elseif (a <= 0.00011)
		tmp = sin(b) * (r / 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, -5.3e-5], t$95$0, If[LessEqual[a, 0.00011], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.3000000000000001e-5 or 1.10000000000000004e-4 < a

   1. Initial program 55.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.8

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

   9. Applied rewrites 55.8%

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

   if -5.3000000000000001e-5 < a < 1.10000000000000004e-4

   1. Initial program 98.9%

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

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 5: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-25}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos b)))))
   (if (<= b -5.8e-7) t_0 (if (<= b 3.6e-25) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(b));
	double tmp;
	if (b <= -5.8e-7) {
		tmp = t_0;
	} else if (b <= 3.6e-25) {
		tmp = b * (r / cos(a));
	} 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(b))
    if (b <= (-5.8d-7)) then
        tmp = t_0
    else if (b <= 3.6d-25) then
        tmp = b * (r / cos(a))
    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(b));
	double tmp;
	if (b <= -5.8e-7) {
		tmp = t_0;
	} else if (b <= 3.6e-25) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(b))
	tmp = 0
	if b <= -5.8e-7:
		tmp = t_0
	elif b <= 3.6e-25:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(b)))
	tmp = 0.0
	if (b <= -5.8e-7)
		tmp = t_0;
	elseif (b <= 3.6e-25)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(b));
	tmp = 0.0;
	if (b <= -5.8e-7)
		tmp = t_0;
	elseif (b <= 3.6e-25)
		tmp = b * (r / cos(a));
	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[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.8e-7], t$95$0, If[LessEqual[b, 3.6e-25], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.7999999999999995e-7 or 3.5999999999999999e-25 < b

   1. Initial program 61.2%

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

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -5.7999999999999995e-7 < b < 3.5999999999999999e-25

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. flip-- N/A

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

      5. cos-diff N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 6: 76.2% 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 79.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 79.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 79.3

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

4. Applied rewrites 79.3%

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

5. Final simplification 79.3%

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

Alternative 7: 51.4% 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 79.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. flip-- N/A

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

   5. cos-diff N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
6. 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 51.3

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

7. Applied rewrites 51.3%

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

Alternative 8: 51.4% accurate, 1.9× speedup

Math

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

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(Float64(b * r) / cos(a))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 51.3

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

5. Applied rewrites 51.3%

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

6. Final simplification 51.3%

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

Alternative 9: 35.1% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 51.3

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

5. Applied rewrites 51.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 37.0%

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

9. Final simplification 37.0%

   \[\leadsto b \cdot r \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))