rsin A (should all be same)

Percentage Accurate: 75.9% → 99.5%

Time: 8.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

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

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 77.4%

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

4. Applied rewrites 99.5%

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

Alternative 2: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -3.05e-5) || !(a <= 6.8e-6))
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	else
		tmp = Float64(Float64(r / cos(b)) * sin(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 = (sin(b) / cos(a)) * r;
	else
		tmp = (r / cos(b)) * sin(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[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.9

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

   7. Applied rewrites 52.9%

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

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

   1. Initial program 97.6%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 77.2%

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

Alternative 3: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-6} \lor \neg \left(b \leq 7 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3e-6) (not (<= b 7e-7)))
   (* (/ r (cos b)) (sin b))
   (* (/ r (cos a)) b)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3e-6) || !(b <= 7e-7)) {
		tmp = (r / cos(b)) * sin(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) :: tmp
    if ((b <= (-3d-6)) .or. (.not. (b <= 7d-7))) then
        tmp = (r / cos(b)) * sin(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 tmp;
	if ((b <= -3e-6) || !(b <= 7e-7)) {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	} else {
		tmp = (r / Math.cos(a)) * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3e-6) or not (b <= 7e-7):
		tmp = (r / math.cos(b)) * math.sin(b)
	else:
		tmp = (r / math.cos(a)) * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -3e-6) || !(b <= 7e-7))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	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 <= -3e-6) || ~((b <= 7e-7)))
		tmp = (r / cos(b)) * sin(b);
	else
		tmp = (r / cos(a)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3e-6], N[Not[LessEqual[b, 7e-7]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.0000000000000001e-6 or 6.99999999999999968e-7 < b

   1. Initial program 59.3%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -3.0000000000000001e-6 < b < 6.99999999999999968e-7

   1. Initial program 99.4%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-6} \lor \neg \left(b \leq 7 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 77.4%

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

Alternative 5: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.3

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

4. Applied rewrites 77.3%

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

Alternative 6: 54.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0064 \lor \neg \left(b \leq 9.5\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0064) (not (<= b 9.5)))
   (/ (* r (sin b)) 1.0)
   (/
    (*
     (fma
      (* (fma 0.008333333333333333 (* b b) -0.16666666666666666) r)
      (* b b)
      r)
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0064) || !(b <= 9.5)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (fma((fma(0.008333333333333333, (b * b), -0.16666666666666666) * r), (b * b), r) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0064) || !(b <= 9.5))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666) * r), Float64(b * b), r) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0064], N[Not[LessEqual[b, 9.5]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * r), $MachinePrecision] * N[(b * b), $MachinePrecision] + r), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00640000000000000031 or 9.5 < b

   1. Initial program 57.9%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.9

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

   5. Applied rewrites 11.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 11.9%

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

   if -0.00640000000000000031 < b < 9.5

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0064 \lor \neg \left(b \leq 9.5\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right) \cdot r, b \cdot b, r\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.6% accurate, 1.7× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $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 58.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 12.0

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

   5. Applied rewrites 12.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 12.3%

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

   if -1.05e14 < b < 1.32e12

   1. Initial program 95.9%

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

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

Alternative 8: 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 77.4%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 9: 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 77.4%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))