rsin A (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 12.4s

Alternatives: 11

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: 76.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. *-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 78.9

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

   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 78.9

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

4. Applied rewrites 78.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 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

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. sin-neg N/A

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

   14. lift-neg.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. *-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 78.9

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

   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 78.9

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

4. Applied rewrites 78.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -1e-5)
   (/ (* r (sin b)) (cos a))
   (if (<= a 2e-8) (* r (/ (sin b) (cos b))) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -1e-5) {
		tmp = (r * sin(b)) / cos(a);
	} else if (a <= 2e-8) {
		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 <= (-1d-5)) then
        tmp = (r * sin(b)) / cos(a)
    else if (a <= 2d-8) 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 <= -1e-5) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else if (a <= 2e-8) {
		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 <= -1e-5:
		tmp = (r * math.sin(b)) / math.cos(a)
	elif a <= 2e-8:
		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 <= -1e-5)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	elseif (a <= 2e-8)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -1e-5)
		tmp = (r * sin(b)) / cos(a);
	elseif (a <= 2e-8)
		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, -1e-5], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e-8], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.00000000000000008e-5

   1. Initial program 51.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   if -1.00000000000000008e-5 < a < 2e-8

   1. Initial program 99.3%

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

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if 2e-8 < a

   1. Initial program 65.0%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 65.1

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

      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 65.1

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

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 64.9

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

   7. Applied rewrites 64.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -1e-5) t_0 (if (<= a 2e-8) (* 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 <= -1e-5) {
		tmp = t_0;
	} else if (a <= 2e-8) {
		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 <= (-1d-5)) then
        tmp = t_0
    else if (a <= 2d-8) 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 <= -1e-5) {
		tmp = t_0;
	} else if (a <= 2e-8) {
		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 <= -1e-5:
		tmp = t_0
	elif a <= 2e-8:
		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 <= -1e-5)
		tmp = t_0;
	elseif (a <= 2e-8)
		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 <= -1e-5)
		tmp = t_0;
	elseif (a <= 2e-8)
		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, -1e-5], t$95$0, If[LessEqual[a, 2e-8], 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 < -1.00000000000000008e-5 or 2e-8 < a

   1. Initial program 58.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. *-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 58.2

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

      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 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -1.00000000000000008e-5 < a < 2e-8

   1. Initial program 99.3%

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

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -4000.0)
     t_0
     (if (<= b 2e-5)
       (/
        (*
         r
         (fma
          (fma
           b
           (* b (fma b (* b -0.0001984126984126984) 0.008333333333333333))
           -0.16666666666666666)
          (* b (* b b))
          b))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(b));
	double tmp;
	if (b <= -4000.0) {
		tmp = t_0;
	} else if (b <= 2e-5) {
		tmp = (r * fma(fma(b, (b * fma(b, (b * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(b)))
	tmp = 0.0
	if (b <= -4000.0)
		tmp = t_0;
	elseif (b <= 2e-5)
		tmp = Float64(Float64(r * fma(fma(b, Float64(b * fma(b, Float64(b * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4000.0], t$95$0, If[LessEqual[b, 2e-5], N[(N[(r * N[(N[(b * N[(b * N[(b * N[(b * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4e3 or 2.00000000000000016e-5 < b

   1. Initial program 57.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 57.7

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

      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 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 57.7

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

   7. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 57.7

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

   9. Applied rewrites 57.7%

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

   if -4e3 < b < 2.00000000000000016e-5

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) + b \cdot 1\right)}}{\cos \left(a + b\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{r \cdot \left(\color{blue}{\left(b \cdot {b}^{2}\right) \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)} + b \cdot 1\right)}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{r \cdot \left(\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right)} + b \cdot 1\right)}{\cos \left(a + b\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right) + \color{blue}{b}\right)}{\cos \left(a + b\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, b \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

   5. Applied rewrites 97.6%

      \[\leadsto \frac{r \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 78.9

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

   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 78.9

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

4. Applied rewrites 78.9%

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

5. Final simplification 78.9%

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

Alternative 7: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 2e-5)
   (/
    (*
     b
     (fma
      (* b b)
      (* r (fma (* b b) 0.008333333333333333 -0.16666666666666666))
      r))
    (cos (+ b a)))
   (* (sin b) (/ r 1.0))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 2e-5) {
		tmp = (b * fma((b * b), (r * fma((b * b), 0.008333333333333333, -0.16666666666666666)), r)) / cos((b + a));
	} else {
		tmp = sin(b) * (r / 1.0);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 2e-5)
		tmp = Float64(Float64(b * fma(Float64(b * b), Float64(r * fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666)), r)) / cos(Float64(b + a)));
	else
		tmp = Float64(sin(b) * Float64(r / 1.0));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 2e-5], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(r * N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000016e-5

   1. Initial program 88.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if 2.00000000000000016e-5 < b

   1. Initial program 51.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 53.1

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

   7. Applied rewrites 53.1%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 11.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 500000000000:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 500000000000.0) (/ (* r b) (cos a)) (* (sin b) (/ r 1.0))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 500000000000.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = sin(b) * (r / 1.0);
	}
	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) :: tmp
    if (b <= 500000000000.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = sin(b) * (r / 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 500000000000.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = Math.sin(b) * (r / 1.0);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 500000000000.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = math.sin(b) * (r / 1.0)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 500000000000.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(sin(b) * Float64(r / 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 500000000000.0)
		tmp = (r * b) / cos(a);
	else
		tmp = sin(b) * (r / 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 500000000000.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e11

   1. Initial program 87.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 68.9

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

   5. Applied rewrites 68.9%

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

   if 5e11 < b

   1. Initial program 50.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 50.4

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

      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 50.4

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 12.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 500000000000:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

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

5. Applied rewrites 53.4%

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

Alternative 10: 51.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

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

5. Applied rewrites 53.4%

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

7. Applied rewrites 53.4%

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

8. Final simplification 53.4%

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

Alternative 11: 35.4% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.6%

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

Reproduce

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