rsin B (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 12.7s

Alternatives: 18

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: 76.3% 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

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

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 79.0

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 79.0

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

4. Applied rewrites 79.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. cos-sum N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))))
   (if (<= t_0 -1e-6)
     (* (sin b) (/ r (cos b)))
     (if (<= t_0 0.05) (* r (/ (sin b) (cos a))) (* r (/ (sin b) (cos b)))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = sin(b) * (r / cos(b));
	} else if (t_0 <= 0.05) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(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 <= (-1d-6)) then
        tmp = sin(b) * (r / cos(b))
    else if (t_0 <= 0.05d0) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(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 <= -1e-6) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (t_0 <= 0.05) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	tmp = 0
	if t_0 <= -1e-6:
		tmp = math.sin(b) * (r / math.cos(b))
	elif t_0 <= 0.05:
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	tmp = 0.0
	if (t_0 <= -1e-6)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (t_0 <= 0.05)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(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 <= -1e-6)
		tmp = sin(b) * (r / cos(b));
	elseif (t_0 <= 0.05)
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(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, -1e-6], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -9.99999999999999955e-7

   1. Initial program 59.2%

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

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-neg N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-cos.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.9

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

   7. Applied rewrites 12.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 12.9

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

   9. Applied rewrites 12.9%

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

   10. Taylor expanded in a around 0

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

       1. lower-cos.f64 59.9

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

   12. Applied rewrites 59.9%

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

   if -9.99999999999999955e-7 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.050000000000000003

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 59.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 59.6

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

   5. Applied rewrites 59.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.05:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))))
   (if (<= t_0 -1e-6)
     (/ (* r (sin b)) (cos b))
     (if (<= t_0 0.05) (* r (/ (sin b) (cos a))) (* r (/ (sin b) (cos b)))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = (r * sin(b)) / cos(b);
	} else if (t_0 <= 0.05) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(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 <= (-1d-6)) then
        tmp = (r * sin(b)) / cos(b)
    else if (t_0 <= 0.05d0) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(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 <= -1e-6) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else if (t_0 <= 0.05) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	tmp = 0
	if t_0 <= -1e-6:
		tmp = (r * math.sin(b)) / math.cos(b)
	elif t_0 <= 0.05:
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	tmp = 0.0
	if (t_0 <= -1e-6)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	elseif (t_0 <= 0.05)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(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 <= -1e-6)
		tmp = (r * sin(b)) / cos(b);
	elseif (t_0 <= 0.05)
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(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, -1e-6], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -9.99999999999999955e-7

   1. Initial program 59.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if -9.99999999999999955e-7 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.050000000000000003

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 59.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 59.6

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

   5. Applied rewrites 59.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.05:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (/ (* r (sin b)) (cos b))))
   (if (<= t_0 -1e-6) t_1 (if (<= t_0 0.05) (* r (/ (sin 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 * sin(b)) / cos(b);
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = r * (sin(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 * sin(b)) / cos(b)
    if (t_0 <= (-1d-6)) then
        tmp = t_1
    else if (t_0 <= 0.05d0) then
        tmp = r * (sin(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.sin(b)) / Math.cos(b);
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = r * (Math.sin(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.sin(b)) / math.cos(b)
	tmp = 0
	if t_0 <= -1e-6:
		tmp = t_1
	elif t_0 <= 0.05:
		tmp = r * (math.sin(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(Float64(r * sin(b)) / cos(b))
	tmp = 0.0
	if (t_0 <= -1e-6)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = Float64(r * Float64(sin(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 * sin(b)) / cos(b);
	tmp = 0.0;
	if (t_0 <= -1e-6)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = r * (sin(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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-6], t$95$1, If[LessEqual[t$95$0, 0.05], N[(r * N[(N[Sin[b], $MachinePrecision] / 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))) < -9.99999999999999955e-7 or 0.050000000000000003 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 59.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if -9.99999999999999955e-7 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.050000000000000003

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.05:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.15:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (/ (* r (sin b)) (cos b))))
   (if (<= t_0 -1e-6) t_1 (if (<= t_0 0.15) (* 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 * sin(b)) / cos(b);
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_1;
	} else if (t_0 <= 0.15) {
		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 * sin(b)) / cos(b)
    if (t_0 <= (-1d-6)) then
        tmp = t_1
    else if (t_0 <= 0.15d0) 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.sin(b)) / Math.cos(b);
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_1;
	} else if (t_0 <= 0.15) {
		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.sin(b)) / math.cos(b)
	tmp = 0
	if t_0 <= -1e-6:
		tmp = t_1
	elif t_0 <= 0.15:
		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(Float64(r * sin(b)) / cos(b))
	tmp = 0.0
	if (t_0 <= -1e-6)
		tmp = t_1;
	elseif (t_0 <= 0.15)
		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 * sin(b)) / cos(b);
	tmp = 0.0;
	if (t_0 <= -1e-6)
		tmp = t_1;
	elseif (t_0 <= 0.15)
		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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-6], t$95$1, If[LessEqual[t$95$0, 0.15], 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))) < -9.99999999999999955e-7 or 0.149999999999999994 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 60.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -9.99999999999999955e-7 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.149999999999999994

   1. Initial program 98.3%

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

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

      2. lower-cos.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.15:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.0%

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

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. sin-neg N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-cos.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 0.4× speedup

Math

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

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 79.0%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 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 79.0%

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

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 76.3% 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 79.0%

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

3. Final simplification 79.0%

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

Alternative 10: 54.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.5:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -4.8)
     t_0
     (if (<= b 4.5)
       (*
        r
        (/
         (fma
          (fma
           b
           (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
           -0.16666666666666666)
          (* b (* b b))
          b)
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -4.8) {
		tmp = t_0;
	} else if (b <= 4.5) {
		tmp = r * (fma(fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -4.8)
		tmp = t_0;
	elseif (b <= 4.5)
		tmp = Float64(r * Float64(fma(fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.8], t$95$0, If[LessEqual[b, 4.5], N[(r * N[(N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 4.5 < b

   1. Initial program 58.3%

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

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-neg N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.5

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.1%

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

   if -4.79999999999999982 < b < 4.5

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 97.6%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 4.5:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.4:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -4.0)
     t_0
     (if (<= b 4.4)
       (*
        r
        (/
         (fma
          (fma (* b b) 0.008333333333333333 -0.16666666666666666)
          (* b (* b b))
          b)
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -4.0) {
		tmp = t_0;
	} else if (b <= 4.4) {
		tmp = r * (fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -4.0)
		tmp = t_0;
	elseif (b <= 4.4)
		tmp = Float64(r * Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.0], t$95$0, If[LessEqual[b, 4.4], N[(r * N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4 or 4.4000000000000004 < b

   1. Initial program 58.3%

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

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-neg N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.5

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.1%

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

   if -4 < b < 4.4000000000000004

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 4.4:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.4:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -4.8)
     t_0
     (if (<= b 2.4)
       (* r (/ (fma b (* b (* b -0.16666666666666666)) b) (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -4.8) {
		tmp = t_0;
	} else if (b <= 2.4) {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -4.8)
		tmp = t_0;
	elseif (b <= 2.4)
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.8], t$95$0, If[LessEqual[b, 2.4], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 2.39999999999999991 < b

   1. Initial program 58.6%

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

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-neg N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.4

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

   7. Applied rewrites 12.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.0%

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

   if -4.79999999999999982 < b < 2.39999999999999991

   1. Initial program 99.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{elif}\;b \leq 2.4:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -3.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 37:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -3.4) t_0 (if (<= b 37.0) (/ (* r b) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -3.4) {
		tmp = t_0;
	} else if (b <= 37.0) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / 1.0d0)
    if (b <= (-3.4d0)) then
        tmp = t_0
    else if (b <= 37.0d0) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / 1.0);
	double tmp;
	if (b <= -3.4) {
		tmp = t_0;
	} else if (b <= 37.0) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / 1.0)
	tmp = 0
	if b <= -3.4:
		tmp = t_0
	elif b <= 37.0:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -3.4)
		tmp = t_0;
	elseif (b <= 37.0)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / 1.0);
	tmp = 0.0;
	if (b <= -3.4)
		tmp = t_0;
	elseif (b <= 37.0)
		tmp = (r * b) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.4], t$95$0, If[LessEqual[b, 37.0], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.39999999999999991 or 37 < b

   1. Initial program 58.3%

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

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-neg N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.5

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.1%

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

   if -3.39999999999999991 < b < 37

   1. Initial program 99.0%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

Alternative 14: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{1}\\ \mathbf{if}\;b \leq -1.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.182:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) 1.0))))
   (if (<= b -1.3) t_0 (if (<= b 0.182) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / 1.0);
	double tmp;
	if (b <= -1.3) {
		tmp = t_0;
	} else if (b <= 0.182) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / 1.0d0)
    if (b <= (-1.3d0)) then
        tmp = t_0
    else if (b <= 0.182d0) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / 1.0);
	double tmp;
	if (b <= -1.3) {
		tmp = t_0;
	} else if (b <= 0.182) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / 1.0)
	tmp = 0
	if b <= -1.3:
		tmp = t_0
	elif b <= 0.182:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / 1.0))
	tmp = 0.0
	if (b <= -1.3)
		tmp = t_0;
	elseif (b <= 0.182)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / 1.0);
	tmp = 0.0;
	if (b <= -1.3)
		tmp = t_0;
	elseif (b <= 0.182)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.3], t$95$0, If[LessEqual[b, 0.182], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.30000000000000004 or 0.182 < b

   1. Initial program 58.7%

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

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-neg N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.4

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

   7. Applied rewrites 12.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 13.1%

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

   if -1.30000000000000004 < b < 0.182

   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. lower-/.f64 N/A

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

      2. lower-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 15: 50.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

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

   2. lower-cos.f64 51.0

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

5. Applied rewrites 51.0%

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

Alternative 16: 50.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 51.0

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

5. Applied rewrites 51.0%

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

7. Applied rewrites 51.0%

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

Alternative 17: 50.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 51.0

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

5. Applied rewrites 51.0%

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

Alternative 18: 33.8% 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 79.0%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))