rsin A (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 9.8s

Alternatives: 10

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.9

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

4. Applied rewrites 77.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-neg.f64 N/A

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

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

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

   17. distribute-lft-neg-out N/A

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

   18. lift-sin.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lift-sin.f64 N/A

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

   21. lower-neg.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -55:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \mathbf{elif}\;b \leq 0.0225:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -55.0)
   (* (/ (sin b) (cos b)) r)
   (if (<= b 0.0225)
     (/
      (*
       (fma
        (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
        (* b b)
        r)
       b)
      (cos (+ a b)))
     (* (/ r (cos b)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -55.0) {
		tmp = (sin(b) / cos(b)) * r;
	} else if (b <= 0.0225) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = (r / cos(b)) * sin(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -55.0)
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	elseif (b <= 0.0225)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -55.0], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[b, 0.0225], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -55

   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. 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 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 56.4

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

   7. Applied rewrites 56.4%

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

   if -55 < b < 0.022499999999999999

   1. Initial program 98.4%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

   if 0.022499999999999999 < b

   1. Initial program 57.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 58.8

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

   5. Applied rewrites 58.8%

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

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0225:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -55.0)
     t_0
     (if (<= b 0.0225)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -55.0) {
		tmp = t_0;
	} else if (b <= 0.0225) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -55.0)
		tmp = t_0;
	elseif (b <= 0.0225)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -55.0], t$95$0, If[LessEqual[b, 0.0225], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -55 or 0.022499999999999999 < b

   1. Initial program 57.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 57.3

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

   5. Applied rewrites 57.3%

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

   if -55 < b < 0.022499999999999999

   1. Initial program 98.4%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

Alternative 5: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.9

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

4. Applied rewrites 77.9%

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

Alternative 6: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1} \cdot \sin b\\ \mathbf{if}\;b \leq -39000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 22:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r 1.0) (sin b))))
   (if (<= b -39000000.0)
     t_0
     (if (<= b 22.0)
       (/
        (*
         (fma
          (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
          (* b b)
          r)
         b)
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / 1.0) * sin(b);
	double tmp;
	if (b <= -39000000.0) {
		tmp = t_0;
	} else if (b <= 22.0) {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / 1.0) * sin(b))
	tmp = 0.0
	if (b <= -39000000.0)
		tmp = t_0;
	elseif (b <= 22.0)
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -39000000.0], t$95$0, If[LessEqual[b, 22.0], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9e7 or 22 < b

   1. Initial program 58.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 58.4

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 57.5

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

   7. Applied rewrites 57.5%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 11.4%

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

   if -3.9e7 < b < 22

   1. Initial program 96.2%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.1%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -39000000:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{elif}\;b \leq 22:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3.3e+14) (* (/ r 1.0) (sin b)) (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = (r / 1.0) * sin(b);
	} else {
		tmp = (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 (b <= (-3.3d+14)) then
        tmp = (r / 1.0d0) * sin(b)
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.3e+14) {
		tmp = (r / 1.0) * Math.sin(b);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3.3e+14:
		tmp = (r / 1.0) * math.sin(b)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3.3e+14)
		tmp = Float64(Float64(r / 1.0) * sin(b));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3.3e+14)
		tmp = (r / 1.0) * sin(b);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3.3e+14], N[(N[(r / 1.0), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3e14

   1. Initial program 56.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 55.1

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 11.9%

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

   if -3.3e14 < b

   1. Initial program 86.4%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.1%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{r}{1} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 50.4

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

5. Applied rewrites 50.4%

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

7. Applied rewrites 50.5%

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

Alternative 9: 51.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 50.4

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

5. Applied rewrites 50.4%

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

7. Applied rewrites 50.5%

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

8. Final simplification 50.5%

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

Alternative 10: 34.9% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.9%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))