rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 14.0s

Alternatives: 12

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: 76.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3700000000 \lor \neg \left(a \leq 5.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -3700000000.0) (not (<= a 5.2e-5)))
   (* (/ (sin b) (cos a)) r)
   (* (/ (sin b) (cos b)) r)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3700000000.0) || !(a <= 5.2e-5)) {
		tmp = (sin(b) / cos(a)) * r;
	} else {
		tmp = (sin(b) / cos(b)) * r;
	}
	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 <= (-3700000000.0d0)) .or. (.not. (a <= 5.2d-5))) then
        tmp = (sin(b) / cos(a)) * r
    else
        tmp = (sin(b) / cos(b)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3700000000.0) || !(a <= 5.2e-5)) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else {
		tmp = (Math.sin(b) / Math.cos(b)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -3700000000.0) or not (a <= 5.2e-5):
		tmp = (math.sin(b) / math.cos(a)) * r
	else:
		tmp = (math.sin(b) / math.cos(b)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -3700000000.0) || !(a <= 5.2e-5))
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	else
		tmp = Float64(Float64(sin(b) / cos(b)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -3700000000.0) || ~((a <= 5.2e-5)))
		tmp = (sin(b) / cos(a)) * r;
	else
		tmp = (sin(b) / cos(b)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -3700000000.0], N[Not[LessEqual[a, 5.2e-5]], $MachinePrecision]], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7e9 or 5.19999999999999968e-5 < a

   1. Initial program 55.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{a \cdot a + \left(b \cdot b - a \cdot b\right)}\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left({a}^{3} + {b}^{3}\right)\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)\right)}} \]

      6. cos-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{\color{blue}{-\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)}}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\left(\left(b \cdot b - a \cdot b\right) + a \cdot a\right)}}\right)} \]

      15. distribute-rgt-out-- N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{b \cdot \left(b - a\right)} + a \cdot a\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{\left(b - a\right) \cdot b} + a \cdot a\right)}\right)} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\mathsf{fma}\left(b - a, b, a \cdot a\right)}}\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(\color{blue}{b - a}, b, a \cdot a\right)}\right)} \]

      19. lower-*.f64 13.5

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, \color{blue}{a \cdot a}\right)}\right)} \]

   4. Applied rewrites 13.5%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      4. sum-cubes N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out-- N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lift-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower--.f64 8.2

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

   6. Applied rewrites 8.2%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 55.8

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

   9. Applied rewrites 55.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]
   10. 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 55.8

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

   11. Applied rewrites 55.8%

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

   if -3.7e9 < a < 5.19999999999999968e-5

   1. Initial program 96.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-cos.f64 96.9

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

   7. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.9

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

   9. Applied rewrites 96.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3700000000 \lor \neg \left(a \leq 5.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos b} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3700000000 \lor \neg \left(a \leq 5.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -3700000000.0) (not (<= a 5.2e-5)))
   (* (/ (sin b) (cos a)) r)
   (* (/ r (cos b)) (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3700000000.0) || !(a <= 5.2e-5)) {
		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 <= (-3700000000.0d0)) .or. (.not. (a <= 5.2d-5))) 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 <= -3700000000.0) || !(a <= 5.2e-5)) {
		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 <= -3700000000.0) or not (a <= 5.2e-5):
		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 <= -3700000000.0) || !(a <= 5.2e-5))
		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 <= -3700000000.0) || ~((a <= 5.2e-5)))
		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, -3700000000.0], N[Not[LessEqual[a, 5.2e-5]], $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 < -3.7e9 or 5.19999999999999968e-5 < a

   1. Initial program 55.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{a \cdot a + \left(b \cdot b - a \cdot b\right)}\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left({a}^{3} + {b}^{3}\right)\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)\right)}} \]

      6. cos-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{\color{blue}{-\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)}}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\left(\left(b \cdot b - a \cdot b\right) + a \cdot a\right)}}\right)} \]

      15. distribute-rgt-out-- N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{b \cdot \left(b - a\right)} + a \cdot a\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{\left(b - a\right) \cdot b} + a \cdot a\right)}\right)} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\mathsf{fma}\left(b - a, b, a \cdot a\right)}}\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(\color{blue}{b - a}, b, a \cdot a\right)}\right)} \]

      19. lower-*.f64 13.5

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, \color{blue}{a \cdot a}\right)}\right)} \]

   4. Applied rewrites 13.5%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      4. sum-cubes N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out-- N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lift-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower--.f64 8.2

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

   6. Applied rewrites 8.2%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 55.8

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

   9. Applied rewrites 55.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]
   10. 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 55.8

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

   11. Applied rewrites 55.8%

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

   if -3.7e9 < a < 5.19999999999999968e-5

   1. Initial program 96.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3700000000 \lor \neg \left(a \leq 5.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3700000000 \lor \neg \left(a \leq 5.2 \cdot 10^{-5}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -3700000000.0) (not (<= a 5.2e-5)))
   (* (sin b) (/ r (cos a)))
   (* (/ r (cos b)) (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3700000000.0) || !(a <= 5.2e-5)) {
		tmp = sin(b) * (r / cos(a));
	} 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 <= (-3700000000.0d0)) .or. (.not. (a <= 5.2d-5))) then
        tmp = sin(b) * (r / cos(a))
    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 <= -3700000000.0) || !(a <= 5.2e-5)) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = (r / Math.cos(b)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -3700000000.0) or not (a <= 5.2e-5):
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = (r / math.cos(b)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -3700000000.0) || !(a <= 5.2e-5))
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	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 <= -3700000000.0) || ~((a <= 5.2e-5)))
		tmp = sin(b) * (r / cos(a));
	else
		tmp = (r / cos(b)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -3700000000.0], N[Not[LessEqual[a, 5.2e-5]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7e9 or 5.19999999999999968e-5 < a

   1. Initial program 55.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{a \cdot a + \left(b \cdot b - a \cdot b\right)}\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left({a}^{3} + {b}^{3}\right)\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)\right)}} \]

      6. cos-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{\color{blue}{-\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)}}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\left(\left(b \cdot b - a \cdot b\right) + a \cdot a\right)}}\right)} \]

      15. distribute-rgt-out-- N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{b \cdot \left(b - a\right)} + a \cdot a\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{\left(b - a\right) \cdot b} + a \cdot a\right)}\right)} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\mathsf{fma}\left(b - a, b, a \cdot a\right)}}\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(\color{blue}{b - a}, b, a \cdot a\right)}\right)} \]

      19. lower-*.f64 13.5

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, \color{blue}{a \cdot a}\right)}\right)} \]

   4. Applied rewrites 13.5%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      4. sum-cubes N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out-- N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lift-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower--.f64 8.2

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

   6. Applied rewrites 8.2%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 55.8

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

   9. Applied rewrites 55.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]
   10. 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. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 55.8

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

   11. Applied rewrites 55.8%

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

   if -3.7e9 < a < 5.19999999999999968e-5

   1. Initial program 96.9%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3700000000 \lor \neg \left(a \leq 5.2 \cdot 10^{-5}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.33 \lor \neg \left(b \leq 0.055\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.33) (not (<= b 0.055)))
   (* (/ r (cos b)) (sin b))
   (/
    (*
     (*
      r
      (fma
       (fma 0.008333333333333333 (* b b) -0.16666666666666666)
       (* b b)
       1.0))
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.33) || !(b <= 0.055)) {
		tmp = (r / cos(b)) * sin(b);
	} else {
		tmp = ((r * fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0)) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.33) || !(b <= 0.055))
		tmp = Float64(Float64(r / cos(b)) * sin(b));
	else
		tmp = Float64(Float64(Float64(r * fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0)) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.33], N[Not[LessEqual[b, 0.055]], $MachinePrecision]], N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[(r * N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.330000000000000016 or 0.0550000000000000003 < b

   1. Initial program 51.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   if -0.330000000000000016 < b < 0.0550000000000000003

   1. Initial program 98.8%

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

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

   5. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.33 \lor \neg \left(b \leq 0.055\right):\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% 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 75.6%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 75.6

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

4. Applied rewrites 75.6%

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

Alternative 8: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 75.6

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

4. Applied rewrites 75.6%

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

Alternative 9: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9 \lor \neg \left(b \leq 2.9\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.9) (not (<= b 2.9)))
   (/ (* r (sin b)) 1.0)
   (/
    (*
     (*
      r
      (fma
       (fma 0.008333333333333333 (* b b) -0.16666666666666666)
       (* b b)
       1.0))
     b)
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.9) || !(b <= 2.9)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = ((r * fma(fma(0.008333333333333333, (b * b), -0.16666666666666666), (b * b), 1.0)) * b) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -3.9) || !(b <= 2.9))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(Float64(r * fma(fma(0.008333333333333333, Float64(b * b), -0.16666666666666666), Float64(b * b), 1.0)) * b) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.9], N[Not[LessEqual[b, 2.9]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(r * N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.89999999999999991 or 2.89999999999999991 < b

   1. Initial program 51.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{a \cdot a + \left(b \cdot b - a \cdot b\right)}\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left({a}^{3} + {b}^{3}\right)\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)\right)}} \]

      6. cos-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{\color{blue}{-\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)}}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\left(\left(b \cdot b - a \cdot b\right) + a \cdot a\right)}}\right)} \]

      15. distribute-rgt-out-- N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{b \cdot \left(b - a\right)} + a \cdot a\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{\left(b - a\right) \cdot b} + a \cdot a\right)}\right)} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\mathsf{fma}\left(b - a, b, a \cdot a\right)}}\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(\color{blue}{b - a}, b, a \cdot a\right)}\right)} \]

      19. lower-*.f64 14.6

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, \color{blue}{a \cdot a}\right)}\right)} \]

   4. Applied rewrites 14.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      4. sum-cubes N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out-- N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lift-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower--.f64 8.1

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

   6. Applied rewrites 8.1%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 11.0

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

   9. Applied rewrites 11.0%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 11.5%

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

   if -3.89999999999999991 < b < 2.89999999999999991

   1. Initial program 98.8%

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

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

   5. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \lor \neg \left(b \leq 2.9\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(r \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, b \cdot b, -0.16666666666666666\right), b \cdot b, 1\right)\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.98 \lor \neg \left(b \leq 4.8\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.98) (not (<= b 4.8)))
   (/ (* r (sin b)) 1.0)
   (* (/ b (cos a)) r)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.98) || !(b <= 4.8)) {
		tmp = (r * sin(b)) / 1.0;
	} else {
		tmp = (b / cos(a)) * r;
	}
	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 <= (-0.98d0)) .or. (.not. (b <= 4.8d0))) then
        tmp = (r * sin(b)) / 1.0d0
    else
        tmp = (b / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.98) || !(b <= 4.8)) {
		tmp = (r * Math.sin(b)) / 1.0;
	} else {
		tmp = (b / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.98) or not (b <= 4.8):
		tmp = (r * math.sin(b)) / 1.0
	else:
		tmp = (b / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.98) || !(b <= 4.8))
		tmp = Float64(Float64(r * sin(b)) / 1.0);
	else
		tmp = Float64(Float64(b / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.98) || ~((b <= 4.8)))
		tmp = (r * sin(b)) / 1.0;
	else
		tmp = (b / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.98], N[Not[LessEqual[b, 4.8]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.97999999999999998 or 4.79999999999999982 < b

   1. Initial program 51.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{a \cdot a + \left(b \cdot b - a \cdot b\right)}\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left({a}^{3} + {b}^{3}\right)\right)}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)\right)}} \]

      6. cos-neg N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(\frac{{a}^{3} + {b}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{\mathsf{neg}\left(\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)\right)}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{\color{blue}{-\left(a \cdot a + \left(b \cdot b - a \cdot b\right)\right)}}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\left(\left(b \cdot b - a \cdot b\right) + a \cdot a\right)}}\right)} \]

      15. distribute-rgt-out-- N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{b \cdot \left(b - a\right)} + a \cdot a\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\left(\color{blue}{\left(b - a\right) \cdot b} + a \cdot a\right)}\right)} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\color{blue}{\mathsf{fma}\left(b - a, b, a \cdot a\right)}}\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(\color{blue}{b - a}, b, a \cdot a\right)}\right)} \]

      19. lower-*.f64 14.6

          \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, \color{blue}{a \cdot a}\right)}\right)} \]

   4. Applied rewrites 14.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(\frac{{b}^{3} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3} + {a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{\color{blue}{{b}^{3}} + {a}^{3}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{r \cdot \sin b}{\cos \left(\frac{{b}^{3} + \color{blue}{{a}^{3}}}{-\mathsf{fma}\left(b - a, b, a \cdot a\right)}\right)} \]

      4. sum-cubes N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-out-- N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lift-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower--.f64 8.1

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

   6. Applied rewrites 8.1%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. cos-neg N/A

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

      3. lower-cos.f64 11.0

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

   9. Applied rewrites 11.0%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 11.5%

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

   if -0.97999999999999998 < b < 4.79999999999999982

   1. Initial program 98.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.98 \lor \neg \left(b \leq 4.8\right):\\ \;\;\;\;\frac{r \cdot \sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 52.4

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

5. Applied rewrites 52.4%

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

7. Applied rewrites 52.5%

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

Alternative 12: 34.8% 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 75.6%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))