rsin B (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 16.9s

Alternatives: 12

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: 76.1% 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

\[\begin{array}{l} \\ \frac{\sin b}{0 - -1} \cdot \frac{r}{\mathsf{fma}\left(\sin b, 0 - \sin a, \cos b \cdot \cos a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (*
  (/ (sin b) (- 0.0 -1.0))
  (/ r (fma (sin b) (- 0.0 (sin a)) (* (cos b) (cos a))))))

C

double code(double r, double a, double b) {
	return (sin(b) / (0.0 - -1.0)) * (r / fma(sin(b), (0.0 - sin(a)), (cos(b) * cos(a))));
}

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) / Float64(0.0 - -1.0)) * Float64(r / fma(sin(b), Float64(0.0 - sin(a)), Float64(cos(b) * cos(a)))))
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] / N[(0.0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(r / N[(N[Sin[b], $MachinePrecision] * N[(0.0 - N[Sin[a], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. neg-sub0 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{0 - \sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   8. --lowering--.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{0 - \sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   9. sin-lowering-sin.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \color{blue}{\sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \color{blue}{\sin a}, \cos a \cdot \cos b\right)} \]

   11. *-lowering-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \color{blue}{\cos a \cdot \cos b}\right)} \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \color{blue}{\cos a} \cdot \cos b\right)} \]

   13. cos-lowering-cos.f64 99.4

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \color{blue}{\cos b}\right)} \]

4. Applied egg-rr 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. cos-lowering-cos.f64 99.4

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

6. Applied egg-rr 99.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

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

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

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

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

   8. *-lowering-*.f64 N/A

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

   9. sub0-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(0 - \sin b\right)} \cdot \sin a\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(0 - \sin b\right)} \cdot \sin a\right)} \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(0 - \color{blue}{\sin b}\right) \cdot \sin a\right)} \]

   12. sin-lowering-sin.f64 99.4

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(0 - \sin b\right) \cdot \color{blue}{\sin a}\right)} \]

8. Applied egg-rr 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(0 - \sin b\right) \cdot \sin a\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin b}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a} \cdot r} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin b\right)}{\mathsf{neg}\left(\left(\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a\right)\right)}} \cdot r \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot r}{\mathsf{neg}\left(\left(\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a\right)\right)}} \]

   4. neg-mul-1 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot r}{\color{blue}{-1 \cdot \left(\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a\right)}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin b\right)}{-1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin b\right)}{-1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin b\right)}{-1}} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a} \]

   8. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{0 - \sin b}}{-1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a} \]

   9. --lowering--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - \sin b}}{-1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a} \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{0 - \color{blue}{\sin b}}{-1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \frac{0 - \sin b}{-1} \cdot \color{blue}{\frac{r}{\cos b \cdot \cos a + \left(0 - \sin b\right) \cdot \sin a}} \]

   12. +-commutative N/A

       \[\leadsto \frac{0 - \sin b}{-1} \cdot \frac{r}{\color{blue}{\left(0 - \sin b\right) \cdot \sin a + \cos b \cdot \cos a}} \]

10. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\frac{0 - \sin b}{-1} \cdot \frac{r}{\mathsf{fma}\left(\sin b, 0 - \sin a, \cos b \cdot \cos a\right)}} \]

11. Final simplification 99.5%

    \[\leadsto \frac{\sin b}{0 - -1} \cdot \frac{r}{\mathsf{fma}\left(\sin b, 0 - \sin a, \cos b \cdot \cos a\right)} \]
12. Add Preprocessing

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))))
   (if (<= t_0 -0.01)
     (/ r (/ 1.0 (tan b)))
     (if (<= t_0 5e-13) (* r (* b (/ 1.0 (cos a)))) (* r (tan b))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = r / (1.0 / tan(b));
	} else if (t_0 <= 5e-13) {
		tmp = r * (b * (1.0 / cos(a)));
	} else {
		tmp = r * tan(b);
	}
	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) / cos((b + a))
    if (t_0 <= (-0.01d0)) then
        tmp = r / (1.0d0 / tan(b))
    else if (t_0 <= 5d-13) then
        tmp = r * (b * (1.0d0 / cos(a)))
    else
        tmp = r * tan(b)
    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 tmp;
	if (t_0 <= -0.01) {
		tmp = r / (1.0 / Math.tan(b));
	} else if (t_0 <= 5e-13) {
		tmp = r * (b * (1.0 / Math.cos(a)));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	tmp = 0
	if t_0 <= -0.01:
		tmp = r / (1.0 / math.tan(b))
	elif t_0 <= 5e-13:
		tmp = r * (b * (1.0 / math.cos(a)))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(r / Float64(1.0 / tan(b)));
	elseif (t_0 <= 5e-13)
		tmp = Float64(r * Float64(b * Float64(1.0 / cos(a))));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((b + a));
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = r / (1.0 / tan(b));
	elseif (t_0 <= 5e-13)
		tmp = r * (b * (1.0 / cos(a)));
	else
		tmp = r * tan(b);
	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]}, If[LessEqual[t$95$0, -0.01], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-13], N[(r * N[(b * N[(1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0100000000000000002

   1. Initial program 69.5%

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

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 69.6

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

   5. Simplified 69.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. clear-num N/A

         \[\leadsto \frac{r}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{r}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}}} \]

      7. quot-tan N/A

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

      8. tan-lowering-tan.f64 69.8

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   7. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   if -0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.9999999999999999e-13

   1. Initial program 99.6%

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

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

      3. /-lowering-/.f64 N/A

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

      4. cos-lowering-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{\cos a}{r}}} \]

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cos-lowering-cos.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   if 4.9999999999999999e-13 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 57.4%

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

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 56.2

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

   5. Simplified 56.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 56.4

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

   7. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\tan b \cdot r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.01:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := r \cdot \tan b\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (* r (tan b))))
   (if (<= t_0 -0.01)
     t_1
     (if (<= t_0 5e-13) (* r (* b (/ 1.0 (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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		tmp = r * (b * (1.0 / 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 <= (-0.01d0)) then
        tmp = t_1
    else if (t_0 <= 5d-13) then
        tmp = r * (b * (1.0d0 / 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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		tmp = r * (b * (1.0 / 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 <= -0.01:
		tmp = t_1
	elif t_0 <= 5e-13:
		tmp = r * (b * (1.0 / 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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		tmp = Float64(r * Float64(b * Float64(1.0 / 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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		tmp = r * (b * (1.0 / 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, -0.01], t$95$1, If[LessEqual[t$95$0, 5e-13], N[(r * N[(b * N[(1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.0100000000000000002 or 4.9999999999999999e-13 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 63.4%

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

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 62.8

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

   5. Simplified 62.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.9999999999999999e-13

   1. Initial program 99.6%

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

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

      3. /-lowering-/.f64 N/A

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

      4. cos-lowering-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{\cos a}{r}}} \]

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cos-lowering-cos.f64 99.6

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

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(b \cdot \frac{1}{\cos a}\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.01:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := r \cdot \tan b\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (* r (tan b))))
   (if (<= t_0 -0.01) t_1 (if (<= t_0 5e-13) (* r (/ b (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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		tmp = r * (b / 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 <= (-0.01d0)) then
        tmp = t_1
    else if (t_0 <= 5d-13) then
        tmp = r * (b / 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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		tmp = r * (b / 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 <= -0.01:
		tmp = t_1
	elif t_0 <= 5e-13:
		tmp = r * (b / 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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		tmp = Float64(r * Float64(b / 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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		tmp = r * (b / 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, -0.01], t$95$1, If[LessEqual[t$95$0, 5e-13], N[(r * N[(b / 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))) < -0.0100000000000000002 or 4.9999999999999999e-13 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 63.4%

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

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 62.8

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

   5. Simplified 62.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.9999999999999999e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cos-lowering-cos.f64 99.6

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

   5. Simplified 99.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.01:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := r \cdot \tan b\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (* r (tan b))))
   (if (<= t_0 -0.01) t_1 (if (<= t_0 5e-13) (* 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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		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 <= (-0.01d0)) then
        tmp = t_1
    else if (t_0 <= 5d-13) 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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-13) {
		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 <= -0.01:
		tmp = t_1
	elif t_0 <= 5e-13:
		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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-13)
		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, -0.01], t$95$1, If[LessEqual[t$95$0, 5e-13], 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))) < -0.0100000000000000002 or 4.9999999999999999e-13 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 63.4%

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

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 62.8

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

   5. Simplified 62.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 4.9999999999999999e-13

   1. Initial program 99.6%

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

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

      3. /-lowering-/.f64 N/A

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

      4. cos-lowering-cos.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.01:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, 0 - \sin a, \cos b \cdot \cos a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (sin b) (- 0.0 (sin a)) (* (cos b) (cos a))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / fma(sin(b), (0.0 - sin(a)), (cos(b) * cos(a))));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(sin(b), Float64(0.0 - sin(a)), Float64(cos(b) * cos(a)))))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] * N[(0.0 - N[Sin[a], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. neg-sub0 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{0 - \sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   8. --lowering--.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{0 - \sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   9. sin-lowering-sin.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \color{blue}{\sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \color{blue}{\sin a}, \cos a \cdot \cos b\right)} \]

   11. *-lowering-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \color{blue}{\cos a \cdot \cos b}\right)} \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \color{blue}{\cos a} \cdot \cos b\right)} \]

   13. cos-lowering-cos.f64 99.4

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \color{blue}{\cos b}\right)} \]

4. Applied egg-rr 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. cos-lowering-cos.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, 0 - \sin a, \cos b \cdot \cos a\right)} \]
8. Add Preprocessing

Alternative 7: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (cos b) (cos a) (* (sin b) (- 0.0 (sin a)))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / fma(cos(b), cos(a), (sin(b) * (0.0 - sin(a)))));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(sin(b) * Float64(0.0 - 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[(0.0 - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. neg-sub0 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{0 - \sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   8. --lowering--.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{0 - \sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   9. sin-lowering-sin.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \color{blue}{\sin b}, \sin a, \cos a \cdot \cos b\right)} \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \color{blue}{\sin a}, \cos a \cdot \cos b\right)} \]

   11. *-lowering-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \color{blue}{\cos a \cdot \cos b}\right)} \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \color{blue}{\cos a} \cdot \cos b\right)} \]

   13. cos-lowering-cos.f64 99.4

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \color{blue}{\cos b}\right)} \]

4. Applied egg-rr 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. cos-lowering-cos.f64 99.4

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

6. Applied egg-rr 99.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

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

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

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

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

   8. *-lowering-*.f64 N/A

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

   9. sub0-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(0 - \sin b\right)} \cdot \sin a\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(0 - \sin b\right)} \cdot \sin a\right)} \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(0 - \color{blue}{\sin b}\right) \cdot \sin a\right)} \]

   12. sin-lowering-sin.f64 99.4

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(0 - \sin b\right) \cdot \color{blue}{\sin a}\right)} \]

8. Applied egg-rr 99.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(0 - \sin b\right) \cdot \sin a\right)}} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

   3. neg-lowering-neg.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. sin-lowering-sin.f64 99.4

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)} \]
12. Add Preprocessing

Alternative 8: 99.5% accurate, 0.4× speedup

Math

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

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

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. --lowering--.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. sin-lowering-sin.f64 99.4

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 9: 76.1% 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 79.5%

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

3. Final simplification 79.5%

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

Alternative 10: 60.0% 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 79.5%

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

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

   2. *-lowering-*.f64 N/A

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

   3. sin-lowering-sin.f64 N/A

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

   4. cos-lowering-cos.f64 60.7

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

5. Simplified 60.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. quot-tan N/A

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

   5. tan-lowering-tan.f64 60.8

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

7. Applied egg-rr 60.8%

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

8. Final simplification 60.8%

   \[\leadsto r \cdot \tan b \]
9. Add Preprocessing

Alternative 11: 34.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\frac{\mathsf{fma}\left(b \cdot b, -0.3333333333333333, 1\right)}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (/ (fma (* b b) -0.3333333333333333 1.0) b)))

C

double code(double r, double a, double b) {
	return r / (fma((b * b), -0.3333333333333333, 1.0) / b);
}

Julia

function code(r, a, b)
	return Float64(r / Float64(fma(Float64(b * b), -0.3333333333333333, 1.0) / b))
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(N[(b * b), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

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

   2. *-lowering-*.f64 N/A

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

   3. sin-lowering-sin.f64 N/A

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

   4. cos-lowering-cos.f64 60.7

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

5. Simplified 60.7%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. clear-num N/A

      \[\leadsto \frac{r}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{r}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}}} \]

   7. quot-tan N/A

      \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   8. tan-lowering-tan.f64 60.7

      \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

7. Applied egg-rr 60.7%

   \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

8. Taylor expanded in b around 0

   \[\leadsto \frac{r}{\color{blue}{\frac{1 + \frac{-1}{3} \cdot {b}^{2}}{b}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{r}{\color{blue}{\frac{1 + \frac{-1}{3} \cdot {b}^{2}}{b}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{r}{\frac{\color{blue}{\frac{-1}{3} \cdot {b}^{2} + 1}}{b}} \]

   3. *-commutative N/A

      \[\leadsto \frac{r}{\frac{\color{blue}{{b}^{2} \cdot \frac{-1}{3}} + 1}{b}} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{r}{\frac{\color{blue}{\mathsf{fma}\left({b}^{2}, \frac{-1}{3}, 1\right)}}{b}} \]

   5. unpow2 N/A

      \[\leadsto \frac{r}{\frac{\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{-1}{3}, 1\right)}{b}} \]

   6. *-lowering-*.f64 29.1

      \[\leadsto \frac{r}{\frac{\mathsf{fma}\left(\color{blue}{b \cdot b}, -0.3333333333333333, 1\right)}{b}} \]

10. Simplified 29.1%

    \[\leadsto \frac{r}{\color{blue}{\frac{\mathsf{fma}\left(b \cdot b, -0.3333333333333333, 1\right)}{b}}} \]
11. Add Preprocessing

Alternative 12: 34.4% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

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

   3. /-lowering-/.f64 N/A

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

   4. cos-lowering-cos.f64 47.1

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

5. Simplified 47.1%

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

6. Taylor expanded in a around 0

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

8. Simplified 28.7%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))