rsin A (should all be same)

Percentage Accurate: 76.0% → 99.5%

Time: 15.1s

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: 76.0% 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

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

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 73.7%

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

   1. associate-/l* 73.8%

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

   2. +-commutative 73.8%

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

3. Simplified 73.8%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-/l* 73.8%

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

   2. +-commutative 73.8%

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

3. Simplified 73.8%

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

   1. clear-num 73.7%

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

   2. un-div-inv 73.7%

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

6. Applied egg-rr 73.7%

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

   1. cos-sum 99.3%

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

   2. div-sub 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in r around 0 99.3%

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

    1. associate-/l* 99.4%

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

    2. fma-neg 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-/l* 73.8%

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

   2. +-commutative 73.8%

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

3. Simplified 73.8%

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

   1. clear-num 73.7%

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

   2. un-div-inv 73.7%

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

6. Applied egg-rr 73.7%

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

   1. cos-sum 99.3%

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

   2. div-sub 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in r around 0 99.3%

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

    1. associate-/l* 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0008 \lor \neg \left(a \leq 7.6 \cdot 10^{-11}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.0008) (not (<= a 7.6e-11)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0008) || !(a <= 7.6e-11)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * tan(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 <= (-0.0008d0)) .or. (.not. (a <= 7.6d-11))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0008) || !(a <= 7.6e-11)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.0008) or not (a <= 7.6e-11):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.0008) || !(a <= 7.6e-11))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.0008) || ~((a <= 7.6e-11)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.0008], N[Not[LessEqual[a, 7.6e-11]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.00000000000000038e-4 or 7.5999999999999996e-11 < a

   1. Initial program 49.5%

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

      1. associate-/l* 49.5%

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

      2. +-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b around 0 49.9%

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

   if -8.00000000000000038e-4 < a < 7.5999999999999996e-11

   1. Initial program 99.6%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 49.6%

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

      2. sqrt-unprod 50.0%

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

      3. pow2 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in a around 0 49.8%

      \[\leadsto r \cdot \sqrt{{\left(\frac{\sin b}{\color{blue}{\cos b}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 49.8%

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

      2. sqrt-pow1 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow1 99.5%

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

      5. quot-tan 99.7%

         \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0008 \lor \neg \left(a \leq 7.6 \cdot 10^{-11}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0078:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0078)
   (* r (/ (sin b) (cos a)))
   (if (<= a 7.6e-11) (* r (tan b)) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0078) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 7.6e-11) {
		tmp = r * tan(b);
	} else {
		tmp = sin(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 (a <= (-0.0078d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 7.6d-11) then
        tmp = r * tan(b)
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0078) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 7.6e-11) {
		tmp = r * Math.tan(b);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.0078:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 7.6e-11:
		tmp = r * math.tan(b)
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0078)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 7.6e-11)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.0078)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 7.6e-11)
		tmp = r * tan(b);
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.0078], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.6e-11], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0077999999999999996

   1. Initial program 50.4%

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

      1. associate-/l* 50.5%

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

      2. +-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in b around 0 51.0%

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

   if -0.0077999999999999996 < a < 7.5999999999999996e-11

   1. Initial program 99.6%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 49.6%

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

      2. sqrt-unprod 50.0%

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

      3. pow2 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in a around 0 49.8%

      \[\leadsto r \cdot \sqrt{{\left(\frac{\sin b}{\color{blue}{\cos b}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 49.8%

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

      2. sqrt-pow1 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow1 99.5%

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

      5. quot-tan 99.7%

         \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 99.7%

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

   11. Simplified 99.7%

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

   if 7.5999999999999996e-11 < a

   1. Initial program 48.1%

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

      1. +-commutative 48.1%

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

   3. Simplified 48.1%

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

      1. associate-*r/ 48.0%

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

      2. add-cube-cbrt 47.5%

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

      3. associate-*l* 47.6%

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

      4. pow2 47.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \]

   6. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)} \]

   7. Taylor expanded in b around 0 48.0%

      \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\color{blue}{\cos a}}\right) \]

   8. Taylor expanded in r around 0 35.5%

      \[\leadsto \color{blue}{\sqrt[3]{{r}^{2}}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos a}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 35.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{{r}^{2}} \cdot \sqrt[3]{r}\right) \cdot \frac{\sin b}{\cos a}} \]

      2. cbrt-prod 26.7%

         \[\leadsto \color{blue}{\sqrt[3]{{r}^{2} \cdot r}} \cdot \frac{\sin b}{\cos a} \]

      3. unpow2 26.7%

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

      4. add-cbrt-cube 48.4%

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

      5. clear-num 48.3%

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

      6. un-div-inv 48.3%

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

   10. Applied egg-rr 48.3%

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

       1. associate-/r/ 48.4%

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

   12. Simplified 48.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0078:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0015:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0015)
   (* r (/ (sin b) (cos a)))
   (if (<= a 7.6e-11) (* r (tan b)) (/ (* r (sin b)) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0015) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 7.6e-11) {
		tmp = r * tan(b);
	} else {
		tmp = (r * sin(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 (a <= (-0.0015d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 7.6d-11) then
        tmp = r * tan(b)
    else
        tmp = (r * sin(b)) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0015) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 7.6e-11) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.0015:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 7.6e-11:
		tmp = r * math.tan(b)
	else:
		tmp = (r * math.sin(b)) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0015)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 7.6e-11)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.0015)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 7.6e-11)
		tmp = r * tan(b);
	else
		tmp = (r * sin(b)) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.0015], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.6e-11], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0015

   1. Initial program 50.4%

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

      1. associate-/l* 50.5%

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

      2. +-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in b around 0 51.0%

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

   if -0.0015 < a < 7.5999999999999996e-11

   1. Initial program 99.6%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 49.6%

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

      2. sqrt-unprod 50.0%

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

      3. pow2 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in a around 0 49.8%

      \[\leadsto r \cdot \sqrt{{\left(\frac{\sin b}{\color{blue}{\cos b}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 49.8%

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

      2. sqrt-pow1 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow1 99.5%

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

      5. quot-tan 99.7%

         \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 99.7%

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

   11. Simplified 99.7%

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

   if 7.5999999999999996e-11 < a

   1. Initial program 48.1%

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

      1. +-commutative 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in b around 0 48.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0015:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% 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 73.7%

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

   1. associate-/l* 73.8%

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

   2. +-commutative 73.8%

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

3. Simplified 73.8%

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

5. Final simplification 73.8%

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

Alternative 8: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-5} \lor \neg \left(b \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5.8e-5) (not (<= b 1.4e-5)))
   (* r (tan b))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5.8e-5) || !(b <= 1.4e-5)) {
		tmp = r * tan(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 <= (-5.8d-5)) .or. (.not. (b <= 1.4d-5))) then
        tmp = r * tan(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 <= -5.8e-5) || !(b <= 1.4e-5)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -5.8e-5) or not (b <= 1.4e-5):
		tmp = r * math.tan(b)
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -5.8e-5) || !(b <= 1.4e-5))
		tmp = Float64(r * tan(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 <= -5.8e-5) || ~((b <= 1.4e-5)))
		tmp = r * tan(b);
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -5.8e-5], N[Not[LessEqual[b, 1.4e-5]], $MachinePrecision]], N[(r * N[Tan[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 < -5.8e-5 or 1.39999999999999998e-5 < b

   1. Initial program 52.0%

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

      1. associate-/l* 52.1%

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

      2. +-commutative 52.1%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 26.3%

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

      2. sqrt-unprod 29.4%

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

      3. pow2 29.4%

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

   6. Applied egg-rr 29.4%

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

   7. Taylor expanded in a around 0 29.4%

      \[\leadsto r \cdot \sqrt{{\left(\frac{\sin b}{\color{blue}{\cos b}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 29.4%

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

      2. sqrt-pow1 52.1%

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

      3. metadata-eval 52.1%

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

      4. pow1 52.1%

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

      5. quot-tan 52.3%

         \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   9. Applied egg-rr 52.3%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 52.3%

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

   11. Simplified 52.3%

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

   if -5.8e-5 < b < 1.39999999999999998e-5

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in b around 0 98.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-5} \lor \neg \left(b \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-5} \lor \neg \left(b \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5.6e-5) (not (<= b 5e-5))) (* r (tan b)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5.6e-5) || !(b <= 5e-5)) {
		tmp = r * tan(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 <= (-5.6d-5)) .or. (.not. (b <= 5d-5))) then
        tmp = r * tan(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 <= -5.6e-5) || !(b <= 5e-5)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -5.6e-5) or not (b <= 5e-5):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -5.6e-5) || !(b <= 5e-5))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -5.6e-5) || ~((b <= 5e-5)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -5.6e-5], N[Not[LessEqual[b, 5e-5]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.59999999999999992e-5 or 5.00000000000000024e-5 < b

   1. Initial program 52.0%

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

      1. associate-/l* 52.1%

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

      2. +-commutative 52.1%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 26.3%

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

      2. sqrt-unprod 29.4%

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

      3. pow2 29.4%

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

   6. Applied egg-rr 29.4%

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

   7. Taylor expanded in a around 0 29.4%

      \[\leadsto r \cdot \sqrt{{\left(\frac{\sin b}{\color{blue}{\cos b}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 29.4%

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

      2. sqrt-pow1 52.1%

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

      3. metadata-eval 52.1%

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

      4. pow1 52.1%

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

      5. quot-tan 52.3%

         \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

   9. Applied egg-rr 52.3%

      \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 52.3%

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

   11. Simplified 52.3%

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

   if -5.59999999999999992e-5 < b < 5.00000000000000024e-5

   1. Initial program 98.7%

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

      1. associate-/l* 98.8%

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

      2. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in b around 0 98.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-5} \lor \neg \left(b \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.5% 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 73.7%

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

   1. +-commutative 73.7%

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

3. Simplified 73.7%

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

   1. associate-*r/ 73.8%

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

   2. add-cube-cbrt 72.6%

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

   3. associate-*l* 72.6%

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

   4. pow2 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \]

6. Applied egg-rr 72.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)} \]

7. Taylor expanded in b around 0 51.3%

   \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\color{blue}{\cos a}}\right) \]

8. Taylor expanded in r around 0 37.1%

   \[\leadsto \color{blue}{\sqrt[3]{{r}^{2}}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos a}\right) \]

9. Taylor expanded in a around 0 37.7%

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

10. Final simplification 37.7%

    \[\leadsto r \cdot \sin b \]
11. Add Preprocessing

Alternative 11: 59.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-/l* 73.8%

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

   2. +-commutative 73.8%

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

3. Simplified 73.8%

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

   1. add-sqr-sqrt 38.4%

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

   2. sqrt-unprod 38.5%

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

   3. pow2 38.5%

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

6. Applied egg-rr 38.5%

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

7. Taylor expanded in a around 0 34.1%

   \[\leadsto r \cdot \sqrt{{\left(\frac{\sin b}{\color{blue}{\cos b}}\right)}^{2}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 34.1%

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

   2. sqrt-pow1 59.7%

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

   3. metadata-eval 59.7%

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

   4. pow1 59.7%

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

   5. quot-tan 59.8%

      \[\leadsto r \cdot \left(1 \cdot \color{blue}{\tan b}\right) \]

9. Applied egg-rr 59.8%

   \[\leadsto r \cdot \color{blue}{\left(1 \cdot \tan b\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 59.8%

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

11. Simplified 59.8%

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

12. Final simplification 59.8%

    \[\leadsto r \cdot \tan b \]
13. Add Preprocessing

Alternative 12: 34.4% 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 73.7%

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

   1. associate-/l* 73.8%

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

   2. +-commutative 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in b around 0 47.7%

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

6. Taylor expanded in a around 0 33.6%

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

7. Final simplification 33.6%

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

Reproduce

herbie shell --seed 2024060 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))