rsin A (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 12.8s

Alternatives: 17

Speedup: 1.0×

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: 77.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. sin-neg N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.7

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

   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 77.7

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

4. Applied rewrites 77.7%

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

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

   12. sin-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   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 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.0025:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0048:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -0.0025)
     t_0
     (if (<= b 0.0048)
       (/ (* b (fma -0.16666666666666666 (* r (* b b)) r)) (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 <= -0.0025) {
		tmp = t_0;
	} else if (b <= 0.0048) {
		tmp = (b * fma(-0.16666666666666666, (r * (b * b)), r)) / 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 <= -0.0025)
		tmp = t_0;
	elseif (b <= 0.0048)
		tmp = Float64(Float64(b * fma(-0.16666666666666666, Float64(r * Float64(b * b)), r)) / 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, -0.0025], t$95$0, If[LessEqual[b, 0.0048], N[(N[(b * N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00250000000000000005 or 0.00479999999999999958 < b

   1. Initial program 62.3%

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

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 62.8

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

   7. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 62.8

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

   9. Applied rewrites 62.8%

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

   if -0.00250000000000000005 < b < 0.00479999999999999958

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0025:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.0048:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -0.0025:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0048:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos b)))))
   (if (<= b -0.0025)
     t_0
     (if (<= b 0.0048)
       (/ (* b (fma -0.16666666666666666 (* r (* b b)) r)) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(b));
	double tmp;
	if (b <= -0.0025) {
		tmp = t_0;
	} else if (b <= 0.0048) {
		tmp = (b * fma(-0.16666666666666666, (r * (b * b)), r)) / cos((b + 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 <= -0.0025)
		tmp = t_0;
	elseif (b <= 0.0048)
		tmp = Float64(Float64(b * fma(-0.16666666666666666, Float64(r * Float64(b * b)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return 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, -0.0025], t$95$0, If[LessEqual[b, 0.0048], N[(N[(b * N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00250000000000000005 or 0.00479999999999999958 < b

   1. Initial program 62.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 62.3

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

      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 62.3

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

   4. Applied rewrites 62.3%

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

   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 62.7

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

   7. Applied rewrites 62.7%

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

   if -0.00250000000000000005 < b < 0.00479999999999999958

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0025:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.0048:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. 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 77.7

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

   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 77.7

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

4. Applied rewrites 77.7%

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

5. Final simplification 77.7%

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

Alternative 7: 77.1% 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 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.7

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

   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 77.7

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

4. Applied rewrites 77.7%

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

5. Final simplification 77.7%

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

Alternative 8: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 77.7

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

   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 77.7

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

4. Applied rewrites 77.7%

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

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 48.6

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

7. Applied rewrites 48.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 48.6

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

9. Applied rewrites 48.6%

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

Alternative 9: 56.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -480000000.0)
     t_0
     (if (<= b 55000000000.0)
       (/
        (*
         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 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -480000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		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(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -480000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		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[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -480000000.0], t$95$0, If[LessEqual[b, 55000000000.0], 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 < -4.8e8 or 5.5e10 < b

   1. Initial program 61.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 61.7

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

      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 61.7

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

   4. Applied rewrites 61.7%

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

   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 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -4.8e8 < b < 5.5e10

   1. Initial program 96.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\cos \left(a + b\right)} \]
   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 92.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b \cdot b, b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -480000000.0)
     t_0
     (if (<= b 55000000000.0)
       (*
        (/ r (cos (+ b a)))
        (fma
         (* b b)
         (*
          b
          (fma
           (* b b)
           (fma b (* b -0.0001984126984126984) 0.008333333333333333)
           -0.16666666666666666))
         b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -480000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = (r / cos((b + a))) * fma((b * b), (b * fma((b * b), fma(b, (b * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666)), b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -480000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(Float64(b * b), Float64(b * fma(Float64(b * b), fma(b, Float64(b * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666)), b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -480000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.8e8 or 5.5e10 < b

   1. Initial program 61.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 61.7

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

      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 61.7

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

   4. Applied rewrites 61.7%

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

   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 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -4.8e8 < b < 5.5e10

   1. Initial program 96.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 92.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -480000000:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b \cdot b, b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -11000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -11000.0)
     t_0
     (if (<= b 1.2e+18)
       (/
        (*
         r
         (*
          b
          (fma
           (* b b)
           (fma (* b b) 0.008333333333333333 -0.16666666666666666)
           1.0)))
        (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -11000.0) {
		tmp = t_0;
	} else if (b <= 1.2e+18) {
		tmp = (r * (b * fma((b * b), fma((b * b), 0.008333333333333333, -0.16666666666666666), 1.0))) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -11000.0)
		tmp = t_0;
	elseif (b <= 1.2e+18)
		tmp = Float64(Float64(r * Float64(b * fma(Float64(b * b), fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), 1.0))) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -11000.0], t$95$0, If[LessEqual[b, 1.2e+18], N[(N[(r * N[(b * N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -11000 or 1.2e18 < b

   1. Initial program 62.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 62.5

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

      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 62.5

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

   4. Applied rewrites 62.5%

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

   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 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -11000 < b < 1.2e18

   1. Initial program 95.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 91.9

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 91.9%

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -11000:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -880000000.0)
     t_0
     (if (<= b 55000000000.0)
       (/ (* b (fma -0.16666666666666666 (* r (* b b)) r)) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = (b * fma(-0.16666666666666666, (r * (b * b)), r)) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(Float64(b * fma(-0.16666666666666666, Float64(r * Float64(b * b)), r)) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(N[(b * N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 5.5e10 < b

   1. Initial program 61.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 61.7

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

      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 61.7

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

   4. Applied rewrites 61.7%

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

   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 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -8.8e8 < b < 5.5e10

   1. Initial program 96.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.16666666666666666, r \cdot \left(b \cdot b\right), r\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -880000000.0)
     t_0
     (if (<= b 55000000000.0)
       (* (/ r (cos (+ b a))) (fma b (* -0.16666666666666666 (* b b)) b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = (r / cos((b + a))) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 5.5e10 < b

   1. Initial program 61.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 61.7

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

      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 61.7

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

   4. Applied rewrites 61.7%

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

   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 11.5

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.4%

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

   if -8.8e8 < b < 5.5e10

   1. Initial program 96.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{1}\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+19}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r 1.0))))
   (if (<= b -880000000.0) t_0 (if (<= b 1.05e+19) (/ (* r b) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 1.05e+19) {
		tmp = (r * b) / 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 / 1.0d0)
    if (b <= (-880000000.0d0)) then
        tmp = t_0
    else if (b <= 1.05d+19) then
        tmp = (r * b) / 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 / 1.0);
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_0;
	} else if (b <= 1.05e+19) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / 1.0)
	tmp = 0
	if b <= -880000000.0:
		tmp = t_0
	elif b <= 1.05e+19:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / 1.0))
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 1.05e+19)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / 1.0);
	tmp = 0.0;
	if (b <= -880000000.0)
		tmp = t_0;
	elseif (b <= 1.05e+19)
		tmp = (r * b) / 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 / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$0, If[LessEqual[b, 1.05e+19], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 1.05e19 < b

   1. Initial program 62.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 62.2

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

      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 62.2

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

   4. Applied rewrites 62.2%

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

   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 11.6

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

   7. Applied rewrites 11.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 11.5%

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

   if -8.8e8 < b < 1.05e19

   1. Initial program 95.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+19}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 44.5

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

5. Applied rewrites 44.5%

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

Alternative 16: 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 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 44.5

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

5. Applied rewrites 44.5%

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

7. Applied rewrites 44.5%

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

8. Final simplification 44.5%

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

Alternative 17: 35.2% 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 77.7%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 44.5

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

5. Applied rewrites 44.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.2%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))