rsin A (should all be same)

Percentage Accurate: 76.0% → 99.4%

Time: 15.2s

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: 76.0% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.5%

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

   2. div-sub 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.5%

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

   2. div-sub 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 99.5%

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

   1. expm1-log1p-u 78.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}\right)\right)} \]

   2. expm1-udef 32.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}\right)} - 1} \]

   3. clear-num 32.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b \cdot \cos a}}} - \sin a}\right)} - 1 \]

   4. *-un-lft-identity 32.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\frac{\color{blue}{1 \cdot \sin b}}{\cos b \cdot \cos a}} - \sin a}\right)} - 1 \]

   5. *-commutative 32.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\frac{1 \cdot \sin b}{\color{blue}{\cos a \cdot \cos b}}} - \sin a}\right)} - 1 \]

   6. times-frac 32.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\color{blue}{\frac{1}{\cos a} \cdot \frac{\sin b}{\cos b}}} - \sin a}\right)} - 1 \]

   7. tan-quot 32.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\frac{1}{\cos a} \cdot \color{blue}{\tan b}} - \sin a}\right)} - 1 \]

8. Applied egg-rr 32.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\frac{1}{\cos a} \cdot \tan b} - \sin a}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 78.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{1}{\frac{1}{\cos a} \cdot \tan b} - \sin a}\right)\right)} \]

   2. expm1-log1p 99.5%

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

   3. associate-/r* 99.5%

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

   4. remove-double-div 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

    \[\leadsto \frac{r}{\frac{\cos a}{\tan b} - \sin a} \]

Alternative 3: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000195 \lor \neg \left(b \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.000195) (not (<= b 4.4e-8)))
   (* (sin b) (/ r (cos b)))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000195) || !(b <= 4.4e-8)) {
		tmp = sin(b) * (r / cos(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 <= (-0.000195d0)) .or. (.not. (b <= 4.4d-8))) then
        tmp = sin(b) * (r / cos(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 <= -0.000195) || !(b <= 4.4e-8)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.000195) or not (b <= 4.4e-8):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.000195) || !(b <= 4.4e-8))
		tmp = Float64(sin(b) * Float64(r / cos(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 <= -0.000195) || ~((b <= 4.4e-8)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.000195], N[Not[LessEqual[b, 4.4e-8]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.94999999999999996e-4 or 4.3999999999999997e-8 < b

   1. Initial program 56.9%

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

      1. associate-/l* 57.0%

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

      2. +-commutative 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in a around 0 56.6%

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

      1. associate-/l* 56.7%

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

      2. associate-/r/ 56.5%

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

   6. Simplified 56.5%

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

   if -1.94999999999999996e-4 < b < 4.3999999999999997e-8

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

      2. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in b around 0 99.3%

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

      1. associate-/r/ 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000195 \lor \neg \left(b \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.88) (not (<= a 26000000.0)))
   (* r (/ (sin b) (cos a)))
   (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (1.0 / ((1.0 / tan(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 ((a <= (-0.88d0)) .or. (.not. (a <= 26000000.0d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.88) or not (a <= 26000000.0):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.88) || !(a <= 26000000.0))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.88) || ~((a <= 26000000.0)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.88], N[Not[LessEqual[a, 26000000.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.880000000000000004 or 2.6e7 < a

   1. Initial program 58.5%

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

      1. associate-/l* 58.6%

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

      2. +-commutative 58.6%

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

   3. Simplified 58.6%

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

      1. clear-num 57.3%

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

      2. associate-/r/ 58.4%

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

      3. clear-num 58.5%

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

   5. Applied egg-rr 58.5%

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

   6. Taylor expanded in b around 0 58.5%

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

   if -0.880000000000000004 < a < 2.6e7

   1. Initial program 97.0%

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

      1. associate-/l* 97.0%

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

      2. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 97.5%

      \[\leadsto \frac{r}{\color{blue}{-1 \cdot a + \frac{\cos b}{\sin b}}} \]
   5. Step-by-step derivation

      1. +-commutative 97.5%

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

      2. mul-1-neg 97.5%

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

      3. unsub-neg 97.5%

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

   6. Simplified 97.5%

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

      1. div-inv 97.5%

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

      2. clear-num 97.5%

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

      3. quot-tan 97.6%

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

   8. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 5: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.88) (not (<= a 26000000.0)))
   (/ r (/ (cos a) (sin b)))
   (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = r / (cos(a) / sin(b));
	} else {
		tmp = r * (1.0 / ((1.0 / tan(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 ((a <= (-0.88d0)) .or. (.not. (a <= 26000000.0d0))) then
        tmp = r / (cos(a) / sin(b))
    else
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.88) or not (a <= 26000000.0):
		tmp = r / (math.cos(a) / math.sin(b))
	else:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.88) || !(a <= 26000000.0))
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	else
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.88) || ~((a <= 26000000.0)))
		tmp = r / (cos(a) / sin(b));
	else
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.88], N[Not[LessEqual[a, 26000000.0]], $MachinePrecision]], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.880000000000000004 or 2.6e7 < a

   1. Initial program 58.5%

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

      1. associate-/l* 58.6%

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

      2. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in b around 0 58.5%

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

   if -0.880000000000000004 < a < 2.6e7

   1. Initial program 97.0%

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

      1. associate-/l* 97.0%

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

      2. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 97.5%

      \[\leadsto \frac{r}{\color{blue}{-1 \cdot a + \frac{\cos b}{\sin b}}} \]
   5. Step-by-step derivation

      1. +-commutative 97.5%

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

      2. mul-1-neg 97.5%

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

      3. unsub-neg 97.5%

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

   6. Simplified 97.5%

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

      1. div-inv 97.5%

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

      2. clear-num 97.5%

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

      3. quot-tan 97.6%

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

   8. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 6: 76.0% 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 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. associate-/r/ 77.8%

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

5. Applied egg-rr 77.8%

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

6. Final simplification 77.8%

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

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 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. clear-num 77.1%

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

   2. associate-/r/ 77.7%

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

   3. clear-num 77.8%

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

5. Applied egg-rr 77.8%

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

6. Final simplification 77.8%

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

Alternative 8: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0014 \lor \neg \left(b \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0014) (not (<= b 4.4e-8)))
   (/ r (- (/ 1.0 (tan b)) a))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0014) || !(b <= 4.4e-8)) {
		tmp = r / ((1.0 / tan(b)) - a);
	} 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 <= (-0.0014d0)) .or. (.not. (b <= 4.4d-8))) then
        tmp = r / ((1.0d0 / tan(b)) - a)
    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 <= -0.0014) || !(b <= 4.4e-8)) {
		tmp = r / ((1.0 / Math.tan(b)) - a);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0014) or not (b <= 4.4e-8):
		tmp = r / ((1.0 / math.tan(b)) - a)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0014) || !(b <= 4.4e-8))
		tmp = Float64(r / Float64(Float64(1.0 / tan(b)) - a));
	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 <= -0.0014) || ~((b <= 4.4e-8)))
		tmp = r / ((1.0 / tan(b)) - a);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0014], N[Not[LessEqual[b, 4.4e-8]], $MachinePrecision]], N[(r / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00139999999999999999 or 4.3999999999999997e-8 < b

   1. Initial program 56.9%

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

      1. associate-/l* 57.0%

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

      2. +-commutative 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in a around 0 54.3%

      \[\leadsto \frac{r}{\color{blue}{-1 \cdot a + \frac{\cos b}{\sin b}}} \]
   5. Step-by-step derivation

      1. +-commutative 54.3%

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

      2. mul-1-neg 54.3%

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

      3. unsub-neg 54.3%

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

   6. Simplified 54.3%

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

      1. expm1-log1p-u 35.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\cos b}{\sin b} - a}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{r}{\frac{\cos b}{\sin b} - a}\right)} - 1} \]

      3. clear-num 13.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}} - a}\right)} - 1 \]

      4. quot-tan 13.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\color{blue}{\tan b}} - a}\right)} - 1 \]

   8. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{r}{\frac{1}{\tan b} - a}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 35.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{1}{\tan b} - a}\right)\right)} \]

      2. expm1-log1p 54.4%

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

   10. Simplified 54.4%

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

   if -0.00139999999999999999 < b < 4.3999999999999997e-8

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

      2. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in b around 0 99.3%

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

      1. associate-/r/ 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0014 \lor \neg \left(b \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 9: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \lor \neg \left(b \leq 76000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6) (not (<= b 76000000000.0)))
   (* r (sin b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6) || !(b <= 76000000000.0)) {
		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 <= (-1.6d0)) .or. (.not. (b <= 76000000000.0d0))) 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 <= -1.6) || !(b <= 76000000000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6) or not (b <= 76000000000.0):
		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 <= -1.6) || !(b <= 76000000000.0))
		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 <= -1.6) || ~((b <= 76000000000.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6], N[Not[LessEqual[b, 76000000000.0]], $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 < -1.6000000000000001 or 7.6e10 < b

   1. Initial program 56.1%

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

      1. +-commutative 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in a around 0 53.6%

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

      1. mul-1-neg 53.6%

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

      2. unsub-neg 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in b around 0 11.5%

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

   if -1.6000000000000001 < b < 7.6e10

   1. Initial program 97.5%

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

      1. associate-/l* 97.5%

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

      2. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in b around 0 95.4%

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

      1. associate-/r/ 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \lor \neg \left(b \leq 76000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 10: 50.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. clear-num 77.1%

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

   2. associate-/r/ 77.7%

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

   3. clear-num 77.8%

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

5. Applied egg-rr 77.8%

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

6. Taylor expanded in b around 0 51.6%

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

7. Final simplification 51.6%

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

Alternative 11: 50.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in b around 0 51.6%

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

   1. associate-/r/ 51.6%

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

6. Applied egg-rr 51.6%

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

7. Final simplification 51.6%

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

Alternative 12: 35.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (+ (* b -0.3333333333333333) (/ 1.0 b))))

C

double code(double r, double a, double b) {
	return r / ((b * -0.3333333333333333) + (1.0 / 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 * (-0.3333333333333333d0)) + (1.0d0 / b))
end function

Java

public static double code(double r, double a, double b) {
	return r / ((b * -0.3333333333333333) + (1.0 / b));
}

Python

def code(r, a, b):
	return r / ((b * -0.3333333333333333) + (1.0 / b))

Julia

function code(r, a, b)
	return Float64(r / Float64(Float64(b * -0.3333333333333333) + Float64(1.0 / b)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r / ((b * -0.3333333333333333) + (1.0 / b));
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(b * -0.3333333333333333), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in b around 0 52.8%

   \[\leadsto \frac{r}{\color{blue}{-1 \cdot \sin a + \left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right)}} \]
5. Step-by-step derivation

   1. +-commutative 52.8%

      \[\leadsto \frac{r}{\color{blue}{\left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right) + -1 \cdot \sin a}} \]

   2. neg-mul-1 52.8%

      \[\leadsto \frac{r}{\left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right) + \color{blue}{\left(-\sin a\right)}} \]

   3. unsub-neg 52.8%

      \[\leadsto \frac{r}{\color{blue}{\left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right) - \sin a}} \]

   4. fma-def 52.8%

      \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, -0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a, \frac{\cos a}{b}\right)} - \sin a} \]

   5. distribute-rgt-out-- 52.8%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \color{blue}{\cos a \cdot \left(-0.5 - -0.16666666666666666\right)}, \frac{\cos a}{b}\right) - \sin a} \]

   6. metadata-eval 52.8%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot \color{blue}{-0.3333333333333333}, \frac{\cos a}{b}\right) - \sin a} \]

6. Simplified 52.8%

   \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \frac{\cos a}{b}\right) - \sin a}} \]

7. Taylor expanded in a around 0 33.5%

   \[\leadsto \color{blue}{\frac{r}{-0.3333333333333333 \cdot b + \frac{1}{b}}} \]

8. Final simplification 33.5%

   \[\leadsto \frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]

Alternative 13: 34.7% 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 77.8%

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

   1. associate-/l* 77.8%

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

   2. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in b around 0 51.6%

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

5. Taylor expanded in a around 0 32.5%

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

   1. *-commutative 32.5%

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

7. Simplified 32.5%

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

8. Final simplification 32.5%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023301 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))