rsin B (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 12.3s

Alternatives: 11

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: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (/ (* r (sin b)) (cos b))))
   (if (<= t_0 -0.002) t_1 (if (<= t_0 2e-11) (* 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 * sin(b)) / cos(b);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_1;
	} else if (t_0 <= 2e-11) {
		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 * sin(b)) / cos(b)
    if (t_0 <= (-0.002d0)) then
        tmp = t_1
    else if (t_0 <= 2d-11) 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.sin(b)) / Math.cos(b);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_1;
	} else if (t_0 <= 2e-11) {
		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.sin(b)) / math.cos(b)
	tmp = 0
	if t_0 <= -0.002:
		tmp = t_1
	elif t_0 <= 2e-11:
		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(Float64(r * sin(b)) / cos(b))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = t_1;
	elseif (t_0 <= 2e-11)
		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 * sin(b)) / cos(b);
	tmp = 0.0;
	if (t_0 <= -0.002)
		tmp = t_1;
	elseif (t_0 <= 2e-11)
		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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], t$95$1, If[LessEqual[t$95$0, 2e-11], 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))) < -2e-3 or 1.99999999999999988e-11 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 58.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.5

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

   5. Applied rewrites 58.5%

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

   if -2e-3 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 1.99999999999999988e-11

   1. Initial program 99.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.002:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 4: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.000135:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0028:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -0.000135) t_0 (if (<= a 0.0028) (* r (/ (sin b) (cos b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -0.000135) {
		tmp = t_0;
	} else if (a <= 0.0028) {
		tmp = r * (sin(b) / cos(b));
	} 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) / cos(a))
    if (a <= (-0.000135d0)) then
        tmp = t_0
    else if (a <= 0.0028d0) then
        tmp = r * (sin(b) / cos(b))
    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) / Math.cos(a));
	double tmp;
	if (a <= -0.000135) {
		tmp = t_0;
	} else if (a <= 0.0028) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -0.000135:
		tmp = t_0
	elif a <= 0.0028:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -0.000135)
		tmp = t_0;
	elseif (a <= 0.0028)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -0.000135)
		tmp = t_0;
	elseif (a <= 0.0028)
		tmp = r * (sin(b) / cos(b));
	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] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.000135], t$95$0, If[LessEqual[a, 0.0028], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000002e-4 or 0.00279999999999999997 < a

   1. Initial program 57.1%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 57.3

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

   5. Applied rewrites 57.3%

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

   if -1.35000000000000002e-4 < a < 0.00279999999999999997

   1. Initial program 98.7%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.000135:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0028:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -0.000135) t_0 (if (<= a 0.0028) (/ (* r (sin b)) (cos b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -0.000135) {
		tmp = t_0;
	} else if (a <= 0.0028) {
		tmp = (r * sin(b)) / cos(b);
	} 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) / cos(a))
    if (a <= (-0.000135d0)) then
        tmp = t_0
    else if (a <= 0.0028d0) then
        tmp = (r * sin(b)) / cos(b)
    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) / Math.cos(a));
	double tmp;
	if (a <= -0.000135) {
		tmp = t_0;
	} else if (a <= 0.0028) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -0.000135:
		tmp = t_0
	elif a <= 0.0028:
		tmp = (r * math.sin(b)) / math.cos(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -0.000135)
		tmp = t_0;
	elseif (a <= 0.0028)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -0.000135)
		tmp = t_0;
	elseif (a <= 0.0028)
		tmp = (r * sin(b)) / cos(b);
	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] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.000135], t$95$0, If[LessEqual[a, 0.0028], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000002e-4 or 0.00279999999999999997 < a

   1. Initial program 57.1%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 57.3

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

   5. Applied rewrites 57.3%

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

   if -1.35000000000000002e-4 < a < 0.00279999999999999997

   1. Initial program 98.7%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 6: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Final simplification 78.8%

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

Alternative 7: 53.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -72:\\ \;\;\;\;r \cdot \frac{\sin b}{\mathsf{fma}\left(a, a \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -72.0)
   (* r (/ (sin b) (fma a (* a -0.5) 1.0)))
   (*
    r
    (/
     (fma
      (fma (* b b) 0.008333333333333333 -0.16666666666666666)
      (* b (* b b))
      b)
     (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -72.0) {
		tmp = r * (sin(b) / fma(a, (a * -0.5), 1.0));
	} else {
		tmp = r * (fma(fma((b * b), 0.008333333333333333, -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -72.0)
		tmp = Float64(r * Float64(sin(b) / fma(a, Float64(a * -0.5), 1.0)));
	else
		tmp = Float64(r * Float64(fma(fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -72.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(a * N[(a * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b < -72

   1. Initial program 64.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 63.6

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, a \cdot \left(-0.5 \cdot \cos b\right) - \sin b, \color{blue}{\cos b}\right)} \]

   5. Applied rewrites 63.6%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(-0.5 \cdot \cos b\right) - \sin b, \cos b\right)}} \]

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 10.4%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \color{blue}{a \cdot -0.5}, 1\right)} \]

   if -72 < b

   1. Initial program 83.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 66.9

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

   5. Applied rewrites 66.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -72:\\ \;\;\;\;r \cdot \frac{\sin b}{\mathsf{fma}\left(a, a \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -30:\\ \;\;\;\;r \cdot \frac{\sin b}{\mathsf{fma}\left(a, a \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -30.0)
   (* r (/ (sin b) (fma a (* a -0.5) 1.0)))
   (* r (/ (fma b (* b (* b -0.16666666666666666)) b) (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -30.0) {
		tmp = r * (sin(b) / fma(a, (a * -0.5), 1.0));
	} else {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / cos((b + a)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -30.0)
		tmp = Float64(r * Float64(sin(b) / fma(a, Float64(a * -0.5), 1.0)));
	else
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / cos(Float64(b + a))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -30.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(a * N[(a * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -30

   1. Initial program 64.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 63.6

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, a \cdot \left(-0.5 \cdot \cos b\right) - \sin b, \color{blue}{\cos b}\right)} \]

   5. Applied rewrites 63.6%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(-0.5 \cdot \cos b\right) - \sin b, \cos b\right)}} \]

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 10.4%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \color{blue}{a \cdot -0.5}, 1\right)} \]

   if -30 < b

   1. Initial program 83.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -30:\\ \;\;\;\;r \cdot \frac{\sin b}{\mathsf{fma}\left(a, a \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 78.7

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

   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 78.7

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

4. Applied rewrites 78.7%

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

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. lower-*.f64 52.0

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

7. Applied rewrites 52.0%

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

8. Final simplification 52.0%

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

Alternative 10: 51.9% 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 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 52.0

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

5. Applied rewrites 52.0%

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

Alternative 11: 34.6% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 52.0

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 34.4%

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

9. Final simplification 34.4%

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

Reproduce

herbie shell --seed 2024225 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))