rsin B (should all be same)

Percentage Accurate: 75.9% → 99.5%

Time: 8.7s

Alternatives: 10

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: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-sin.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

Alternative 2: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(a + b\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-6} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ a b)))))
   (if (or (<= t_0 -2e-6) (not (<= t_0 2e-8)))
     (* (sin b) (/ r (cos b)))
     (* (/ r (cos a)) b))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((a + b));
	double tmp;
	if ((t_0 <= -2e-6) || !(t_0 <= 2e-8)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (r / cos(a)) * 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((a + b))
    if ((t_0 <= (-2d-6)) .or. (.not. (t_0 <= 2d-8))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (r / cos(a)) * 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((a + b));
	double tmp;
	if ((t_0 <= -2e-6) || !(t_0 <= 2e-8)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (r / Math.cos(a)) * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((a + b))
	tmp = 0
	if (t_0 <= -2e-6) or not (t_0 <= 2e-8):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (r / math.cos(a)) * b
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(a + b)))
	tmp = 0.0
	if ((t_0 <= -2e-6) || !(t_0 <= 2e-8))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(r / cos(a)) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((a + b));
	tmp = 0.0;
	if ((t_0 <= -2e-6) || ~((t_0 <= 2e-8)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (r / cos(a)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-6], N[Not[LessEqual[t$95$0, 2e-8]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -1.99999999999999991e-6 or 2e-8 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 59.3%

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

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -1.99999999999999991e-6 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 2e-8

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -2 \cdot 10^{-6} \lor \neg \left(\frac{\sin b}{\cos \left(a + b\right)} \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{-5} \lor \neg \left(a \leq 6.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -3.05e-5) (not (<= a 6.8e-6)))
   (* r (/ (sin b) (cos a)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3.05e-5) || !(a <= 6.8e-6)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = sin(b) * (r / 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) :: tmp
    if ((a <= (-3.05d-5)) .or. (.not. (a <= 6.8d-6))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3.05e-5) || !(a <= 6.8e-6)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -3.05e-5) or not (a <= 6.8e-6):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -3.05e-5) || !(a <= 6.8e-6))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -3.05e-5) || ~((a <= 6.8e-6)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -3.05e-5], N[Not[LessEqual[a, 6.8e-6]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.04999999999999994e-5 or 6.80000000000000012e-6 < a

   1. Initial program 53.4%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if -3.04999999999999994e-5 < a < 6.80000000000000012e-6

   1. Initial program 97.5%

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

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{-5} \lor \neg \left(a \leq 6.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

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

   6. lower-*.f64 77.4

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

4. Applied rewrites 77.4%

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

Alternative 5: 75.9% 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]

Derivation

1. Initial program 77.3%

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

Alternative 6: 54.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2050000000000 \lor \neg \left(b \leq 4.9\right):\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, b, 0\right), b, 1\right), b, 0\right)}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2050000000000.0) (not (<= b 4.9)))
   (* r (/ (sin b) 1.0))
   (*
    r
    (/
     (fma (fma (fma -0.16666666666666666 b 0.0) b 1.0) b 0.0)
     (cos (+ a b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2050000000000.0) || !(b <= 4.9)) {
		tmp = r * (sin(b) / 1.0);
	} else {
		tmp = r * (fma(fma(fma(-0.16666666666666666, b, 0.0), b, 1.0), b, 0.0) / cos((a + b)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -2050000000000.0) || !(b <= 4.9))
		tmp = Float64(r * Float64(sin(b) / 1.0));
	else
		tmp = Float64(r * Float64(fma(fma(fma(-0.16666666666666666, b, 0.0), b, 1.0), b, 0.0) / cos(Float64(a + b))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2050000000000.0], N[Not[LessEqual[b, 4.9]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[(N[(N[(-0.16666666666666666 * b + 0.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 0.0), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.05e12 or 4.9000000000000004 < b

   1. Initial program 58.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 12.1%

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

   if -2.05e12 < b < 4.9000000000000004

   1. Initial program 97.9%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-+PI-rev N/A

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

      4. sin-neg-rev N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-PI.f64 10.7

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

   4. Applied rewrites 10.7%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 96.1%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, b, 0\right), b, 1\right), b, 0\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2050000000000 \lor \neg \left(b \leq 4.9\right):\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, b, 0\right), b, 1\right), b, 0\right)}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0064 \lor \neg \left(b \leq 1.65 \cdot 10^{+15}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0064) (not (<= b 1.65e+15)))
   (* r (/ (sin b) 1.0))
   (* r (/ b (cos (+ a b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0064) || !(b <= 1.65e+15)) {
		tmp = r * (sin(b) / 1.0);
	} else {
		tmp = r * (b / cos((a + 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) :: tmp
    if ((b <= (-0.0064d0)) .or. (.not. (b <= 1.65d+15))) then
        tmp = r * (sin(b) / 1.0d0)
    else
        tmp = r * (b / cos((a + b)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0064) || !(b <= 1.65e+15)) {
		tmp = r * (Math.sin(b) / 1.0);
	} else {
		tmp = r * (b / Math.cos((a + b)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0064) or not (b <= 1.65e+15):
		tmp = r * (math.sin(b) / 1.0)
	else:
		tmp = r * (b / math.cos((a + b)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0064) || !(b <= 1.65e+15))
		tmp = Float64(r * Float64(sin(b) / 1.0));
	else
		tmp = Float64(r * Float64(b / cos(Float64(a + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0064) || ~((b <= 1.65e+15)))
		tmp = r * (sin(b) / 1.0);
	else
		tmp = r * (b / cos((a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0064], N[Not[LessEqual[b, 1.65e+15]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00640000000000000031 or 1.65e15 < b

   1. Initial program 57.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 6.9

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

   5. Applied rewrites 6.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 12.2%

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

   if -0.00640000000000000031 < b < 1.65e15

   1. Initial program 98.2%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-+PI-rev N/A

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

      4. sin-neg-rev N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-PI.f64 11.4

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

   4. Applied rewrites 11.4%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. cos-neg N/A

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

      4. cos-PI N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f64 N/A

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

      9. sin-neg N/A

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

      10. sin-PI N/A

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

      11. metadata-eval 94.5

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

   7. Applied rewrites 94.5%

      \[\leadsto r \cdot \frac{\color{blue}{0 + b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0064 \lor \neg \left(b \leq 1.65 \cdot 10^{+15}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+14} \lor \neg \left(b \leq 1320000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.05e+14) (not (<= b 1320000000000.0)))
   (* r (/ (sin b) 1.0))
   (* (/ r (cos a)) b)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.05e+14) || !(b <= 1320000000000.0)) {
		tmp = r * (sin(b) / 1.0);
	} else {
		tmp = (r / cos(a)) * 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) :: tmp
    if ((b <= (-1.05d+14)) .or. (.not. (b <= 1320000000000.0d0))) then
        tmp = r * (sin(b) / 1.0d0)
    else
        tmp = (r / cos(a)) * b
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.05e+14) || !(b <= 1320000000000.0)) {
		tmp = r * (Math.sin(b) / 1.0);
	} else {
		tmp = (r / Math.cos(a)) * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.05e+14) or not (b <= 1320000000000.0):
		tmp = r * (math.sin(b) / 1.0)
	else:
		tmp = (r / math.cos(a)) * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.05e+14) || !(b <= 1320000000000.0))
		tmp = Float64(r * Float64(sin(b) / 1.0));
	else
		tmp = Float64(Float64(r / cos(a)) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.05e+14) || ~((b <= 1320000000000.0)))
		tmp = r * (sin(b) / 1.0);
	else
		tmp = (r / cos(a)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.05e+14], N[Not[LessEqual[b, 1320000000000.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e14 or 1.32e12 < b

   1. Initial program 58.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 6.5

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 12.3%

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

   if -1.05e14 < b < 1.32e12

   1. Initial program 96.0%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+14} \lor \neg \left(b \leq 1320000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 48.0

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

5. Applied rewrites 48.0%

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

Alternative 10: 33.9% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 48.0

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

5. Applied rewrites 48.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 32.6%

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

Reproduce

herbie shell --seed 2024343 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))