rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 12.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / 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 78.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(b + a\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := r \cdot \tan b\\ \mathbf{if}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{r}{t\_0} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ b a))) (t_1 (/ (sin b) t_0)) (t_2 (* r (tan b))))
   (if (<= t_1 -0.04)
     t_2
     (if (<= t_1 0.0002)
       (* (/ r t_0) (fma b (* -0.16666666666666666 (* b b)) b))
       t_2))))

C

double code(double r, double a, double b) {
	double t_0 = cos((b + a));
	double t_1 = sin(b) / t_0;
	double t_2 = r * tan(b);
	double tmp;
	if (t_1 <= -0.04) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = (r / t_0) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = cos(Float64(b + a))
	t_1 = Float64(sin(b) / t_0)
	t_2 = Float64(r * tan(b))
	tmp = 0.0
	if (t_1 <= -0.04)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(r / t_0) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.04], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(r / t$95$0), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0400000000000000008 or 2.0000000000000001e-4 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 60.7

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.7%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 2.0000000000000001e-4

   1. Initial program 97.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

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

   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 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.04:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.0002:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(b + a\right)\\ t_1 := \frac{\sin b}{t\_0}\\ t_2 := r \cdot \tan b\\ \mathbf{if}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ b a))) (t_1 (/ (sin b) t_0)) (t_2 (* r (tan b))))
   (if (<= t_1 -0.04)
     t_2
     (if (<= t_1 0.0002)
       (* r (/ (fma b (* b (* b -0.16666666666666666)) b) t_0))
       t_2))))

C

double code(double r, double a, double b) {
	double t_0 = cos((b + a));
	double t_1 = sin(b) / t_0;
	double t_2 = r * tan(b);
	double tmp;
	if (t_1 <= -0.04) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / t_0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = cos(Float64(b + a))
	t_1 = Float64(sin(b) / t_0)
	t_2 = Float64(r * tan(b))
	tmp = 0.0
	if (t_1 <= -0.04)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / t_0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.04], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0400000000000000008 or 2.0000000000000001e-4 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 60.7

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.7%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 2.0000000000000001e-4

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.04:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.0002:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := r \cdot \tan b\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (* r (tan b))))
   (if (<= t_0 -0.04) t_1 (if (<= t_0 0.0002) (* b (/ r (cos a))) t_1))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double t_1 = r * tan(b);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.0002) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(b) / cos((b + a))
    t_1 = r * tan(b)
    if (t_0 <= (-0.04d0)) then
        tmp = t_1
    else if (t_0 <= 0.0002d0) then
        tmp = b * (r / cos(a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) / Math.cos((b + a));
	double t_1 = r * Math.tan(b);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.0002) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	t_1 = r * math.tan(b)
	tmp = 0
	if t_0 <= -0.04:
		tmp = t_1
	elif t_0 <= 0.0002:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_1
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	t_1 = Float64(r * tan(b))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.0002)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((b + a));
	t_1 = r * tan(b);
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.0002)
		tmp = b * (r / cos(a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.0002], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0400000000000000008 or 2.0000000000000001e-4 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 60.7

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.7%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 2.0000000000000001e-4

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.04:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.0002:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 6: 76.8% 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 78.0%

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

3. Final simplification 78.0%

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

Alternative 7: 60.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return r * tan(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 * tan(b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 77.9

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

4. Applied rewrites 77.9%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 64.1

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

7. Applied rewrites 64.1%

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

9. Applied rewrites 64.2%

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

Alternative 8: 34.3% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 47.8

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

5. Applied rewrites 47.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 34.0%

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

9. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2024232 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))