rsin A (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 14.1s

Alternatives: 13

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: 76.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. add-exp-log 42.2%

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

6. Applied egg-rr 42.2%

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

   1. rem-exp-log 75.1%

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

   2. div-inv 75.0%

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

   3. *-commutative 75.0%

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

   4. associate-*r* 74.9%

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

   5. associate-*l/ 75.0%

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

   6. *-un-lft-identity 75.0%

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

8. Applied egg-rr 75.0%

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

   1. cos-sum 99.5%

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.0%

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

   2. remove-double-neg 75.0%

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

   3. remove-double-neg 75.0%

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

   4. +-commutative 75.0%

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

3. Simplified 75.0%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 4: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* (sin b) (/ r (fma (cos b) (cos a) 0.0))))

C

double code(double r, double a, double b) {
	return sin(b) * (r / fma(cos(b), cos(a), 0.0));
}

Julia

function code(r, a, b)
	return Float64(sin(b) * Float64(r / fma(cos(b), cos(a), 0.0)))
end

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. add-exp-log 42.2%

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

6. Applied egg-rr 42.2%

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

   1. rem-exp-log 75.1%

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

   2. div-inv 75.0%

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

   3. *-commutative 75.0%

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

   4. associate-*r* 74.9%

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

   5. associate-*l/ 75.0%

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

   6. *-un-lft-identity 75.0%

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

8. Applied egg-rr 75.0%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

10. Applied egg-rr 99.5%

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

    1. add-sqr-sqrt 50.1%

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

    2. sqrt-unprod 85.7%

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

    3. sqr-neg 85.7%

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

    4. sqrt-unprod 35.5%

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

    5. add-sqr-sqrt 75.1%

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

    6. sin-mult 76.6%

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

    7. cos-diff 75.6%

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

    8. add-sqr-sqrt 35.9%

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

    9. sqrt-unprod 76.2%

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

    10. sqr-neg 76.2%

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

    11. sqrt-unprod 40.3%

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

    12. add-sqr-sqrt 77.2%

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

    13. distribute-lft-neg-out 77.2%

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

    14. unsub-neg 77.2%

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

    15. cos-sum 76.4%

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

12. Applied egg-rr 76.4%

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

    1. +-inverses 76.4%

       \[\leadsto \frac{r}{\mathsf{fma}\left(\cos b, \cos a, \frac{\color{blue}{0}}{2}\right)} \cdot \sin b \]

    2. metadata-eval 76.4%

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

14. Simplified 76.4%

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

15. Final simplification 76.4%

    \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]
16. Add Preprocessing

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-5} \lor \neg \left(b \leq 2100\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.1e-5) (not (<= b 2100.0)))
   (/ (* r (sin b)) (cos b))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.1e-5) || !(b <= 2100.0)) {
		tmp = (r * sin(b)) / cos(b);
	} else {
		tmp = (r * b) / cos((b + a));
	}
	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 <= (-4.1d-5)) .or. (.not. (b <= 2100.0d0))) then
        tmp = (r * sin(b)) / cos(b)
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.1e-5) || !(b <= 2100.0)) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.1e-5) or not (b <= 2100.0):
		tmp = (r * math.sin(b)) / math.cos(b)
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.1e-5) || !(b <= 2100.0))
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.1e-5) || ~((b <= 2100.0)))
		tmp = (r * sin(b)) / cos(b);
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.1e-5], N[Not[LessEqual[b, 2100.0]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.10000000000000005e-5 or 2100 < b

   1. Initial program 47.8%

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

      1. associate-/l* 47.8%

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

      2. remove-double-neg 47.8%

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

      3. remove-double-neg 47.8%

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

      4. +-commutative 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in a around 0 48.3%

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

   if -4.10000000000000005e-5 < b < 2100

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in b around 0 97.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-5} \lor \neg \left(b \leq 2100\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000112 \lor \neg \left(b \leq 2100\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.000112) (not (<= b 2100.0)))
   (* r (/ (sin b) (cos b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000112) || !(b <= 2100.0)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (r * b) / cos((b + a));
	}
	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 <= (-0.000112d0)) .or. (.not. (b <= 2100.0d0))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000112) || !(b <= 2100.0)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.000112) or not (b <= 2100.0):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.000112) || !(b <= 2100.0))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.000112) || ~((b <= 2100.0)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.000112], N[Not[LessEqual[b, 2100.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.11999999999999998e-4 or 2100 < b

   1. Initial program 47.8%

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

      1. associate-/l* 47.8%

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

      2. remove-double-neg 47.8%

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

      3. remove-double-neg 47.8%

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

      4. +-commutative 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in a around 0 48.3%

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

      1. associate-/l* 48.3%

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

   7. Simplified 48.3%

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

   if -1.11999999999999998e-4 < b < 2100

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in b around 0 97.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000112 \lor \neg \left(b \leq 2100\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. add-exp-log 42.2%

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

6. Applied egg-rr 42.2%

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

   1. rem-exp-log 75.1%

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

   2. div-inv 75.0%

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

   3. *-commutative 75.0%

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

   4. associate-*r* 74.9%

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

   5. associate-*l/ 75.0%

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

   6. *-un-lft-identity 75.0%

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

8. Applied egg-rr 75.0%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

10. Applied egg-rr 99.5%

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

    1. fma-undefine 99.5%

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

    2. add-sqr-sqrt 50.1%

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

    3. sqrt-unprod 85.7%

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

    4. sqr-neg 85.7%

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

    5. sqrt-unprod 35.5%

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

    6. add-sqr-sqrt 75.1%

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

    7. cos-diff 75.2%

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

12. Applied egg-rr 75.2%

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

13. Final simplification 75.2%

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

Alternative 8: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.0%

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

   2. remove-double-neg 75.0%

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

   3. remove-double-neg 75.0%

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

   4. +-commutative 75.0%

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

3. Simplified 75.0%

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

Alternative 9: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.0%

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

   2. remove-double-neg 75.0%

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

   3. remove-double-neg 75.0%

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

   4. +-commutative 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in b around 0 58.8%

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

Alternative 10: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -24 \lor \neg \left(b \leq 1.3 \cdot 10^{+31}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -24.0) (not (<= b 1.3e+31))) (* r (sin b)) (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -24.0) || !(b <= 1.3e+31)) {
		tmp = r * sin(b);
	} else {
		tmp = (r * b) / cos(a);
	}
	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 <= (-24.0d0)) .or. (.not. (b <= 1.3d+31))) then
        tmp = r * sin(b)
    else
        tmp = (r * b) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -24.0) || !(b <= 1.3e+31)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -24.0) or not (b <= 1.3e+31):
		tmp = r * math.sin(b)
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -24.0) || !(b <= 1.3e+31))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -24.0) || ~((b <= 1.3e+31)))
		tmp = r * sin(b);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -24.0], N[Not[LessEqual[b, 1.3e+31]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -24 or 1.3e31 < b

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

   3. Simplified 48.3%

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

      1. clear-num 48.2%

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

      2. associate-/r/ 48.3%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in b around 0 11.5%

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

   8. Taylor expanded in a around 0 13.8%

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

   if -24 < b < 1.3e31

   1. Initial program 95.3%

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

      1. associate-/l* 95.2%

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

      2. remove-double-neg 95.2%

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

      3. remove-double-neg 95.2%

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

      4. +-commutative 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in b around 0 94.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -24 \lor \neg \left(b \leq 1.3 \cdot 10^{+31}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -24 \lor \neg \left(b \leq 1.3 \cdot 10^{+31}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -24.0) (not (<= b 1.3e+31))) (* r (sin b)) (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -24.0) || !(b <= 1.3e+31)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos(a));
	}
	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 <= (-24.0d0)) .or. (.not. (b <= 1.3d+31))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -24.0) || !(b <= 1.3e+31)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -24.0) or not (b <= 1.3e+31):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -24.0) || !(b <= 1.3e+31))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -24.0) || ~((b <= 1.3e+31)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -24.0], N[Not[LessEqual[b, 1.3e+31]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -24 or 1.3e31 < b

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

   3. Simplified 48.3%

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

      1. clear-num 48.2%

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

      2. associate-/r/ 48.3%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in b around 0 11.5%

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

   8. Taylor expanded in a around 0 13.8%

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

   if -24 < b < 1.3e31

   1. Initial program 95.3%

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

      1. associate-/l* 95.2%

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

      2. remove-double-neg 95.2%

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

      3. remove-double-neg 95.2%

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

      4. +-commutative 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in b around 0 94.6%

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

      1. associate-/l* 94.6%

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

   7. Simplified 94.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -24 \lor \neg \left(b \leq 1.3 \cdot 10^{+31}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. clear-num 74.3%

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

   2. associate-/r/ 75.0%

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

6. Applied egg-rr 75.0%

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

7. Taylor expanded in b around 0 58.8%

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

8. Taylor expanded in a around 0 41.7%

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

Alternative 13: 35.1% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.0%

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

   2. remove-double-neg 75.0%

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

   3. remove-double-neg 75.0%

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

   4. +-commutative 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in b around 0 55.6%

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

   1. associate-/l* 55.5%

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

7. Simplified 55.5%

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

8. Taylor expanded in a around 0 37.6%

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

9. Final simplification 37.6%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))