rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 14.5s

Alternatives: 14

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 75.4%

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

   2. +-commutative 75.4%

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

3. Simplified 75.4%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.5%

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

   2. *-commutative 75.5%

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

   3. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0039 \lor \neg \left(a \leq 61000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.0039) (not (<= a 61000000000.0)))
   (* 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 <= -0.0039) || !(a <= 61000000000.0)) {
		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 <= (-0.0039d0)) .or. (.not. (a <= 61000000000.0d0))) 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 <= -0.0039) || !(a <= 61000000000.0)) {
		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 <= -0.0039) or not (a <= 61000000000.0):
		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 <= -0.0039) || !(a <= 61000000000.0))
		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 <= -0.0039) || ~((a <= 61000000000.0)))
		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, -0.0039], N[Not[LessEqual[a, 61000000000.0]], $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 < -0.0038999999999999998 or 6.1e10 < a

   1. Initial program 51.8%

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

      1. associate-*r/ 51.8%

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

      2. *-commutative 51.8%

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

      3. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in b around 0 52.0%

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

   if -0.0038999999999999998 < a < 6.1e10

   1. Initial program 96.3%

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

      1. associate-/l* 96.2%

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

      2. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in a around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   6. Simplified 96.9%

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

      1. div-inv 97.0%

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

      2. clear-num 96.9%

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

      3. quot-tan 97.1%

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

   8. Applied egg-rr 97.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0039 \lor \neg \left(a \leq 61000000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0022:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 61000000000:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0022)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 61000000000.0)
     (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))
     (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0022) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 61000000000.0) {
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	} 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 <= (-0.0022d0)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 61000000000.0d0) then
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    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 <= -0.0022) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 61000000000.0) {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.0022:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 61000000000.0:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0022)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 61000000000.0)
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.0022)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 61000000000.0)
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.0022], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 61000000000.0], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.00220000000000000013

   1. Initial program 48.7%

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

      1. associate-/l* 48.8%

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

      2. +-commutative 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in b around 0 49.7%

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

   if -0.00220000000000000013 < a < 6.1e10

   1. Initial program 96.3%

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

      1. associate-/l* 96.2%

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

      2. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in a around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   6. Simplified 96.9%

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

      1. div-inv 97.0%

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

      2. clear-num 96.9%

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

      3. quot-tan 97.1%

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

   8. Applied egg-rr 97.1%

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

   if 6.1e10 < a

   1. Initial program 54.1%

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

      1. associate-*r/ 54.0%

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

      2. *-commutative 54.0%

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

      3. +-commutative 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in b around 0 53.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0022:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 61000000000:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000165:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 61000000000:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000165)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 61000000000.0)
     (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))
     (/ (* r (sin b)) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000165) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 61000000000.0) {
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	} 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 <= (-0.000165d0)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 61000000000.0d0) then
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    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 <= -0.000165) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 61000000000.0) {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000165:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 61000000000.0:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	else:
		tmp = (r * math.sin(b)) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000165)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 61000000000.0)
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	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 <= -0.000165)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 61000000000.0)
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	else
		tmp = (r * sin(b)) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000165], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 61000000000.0], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $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 < -1.65e-4

   1. Initial program 48.7%

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

      1. associate-/l* 48.8%

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

      2. +-commutative 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in b around 0 49.7%

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

   if -1.65e-4 < a < 6.1e10

   1. Initial program 96.3%

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

      1. associate-/l* 96.2%

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

      2. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in a around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   6. Simplified 96.9%

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

      1. div-inv 97.0%

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

      2. clear-num 96.9%

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

      3. quot-tan 97.1%

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

   8. Applied egg-rr 97.1%

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

   if 6.1e10 < a

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in b around 0 53.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000165:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 61000000000:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]

Alternative 8: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 75.4%

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

   2. +-commutative 75.4%

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

3. Simplified 75.4%

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

   1. associate-/r/ 75.5%

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

5. Applied egg-rr 75.5%

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

6. Final simplification 75.5%

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

Alternative 9: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000106\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00014) (not (<= b 0.000106)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 0.000106)) {
		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 <= (-0.00014d0)) .or. (.not. (b <= 0.000106d0))) 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 <= -0.00014) || !(b <= 0.000106)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00014) or not (b <= 0.000106):
		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 <= -0.00014) || !(b <= 0.000106))
		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 <= -0.00014) || ~((b <= 0.000106)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00014], N[Not[LessEqual[b, 0.000106]], $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 < -1.3999999999999999e-4 or 1.06e-4 < b

   1. Initial program 51.4%

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

      1. associate-*r/ 51.4%

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

      2. *-commutative 51.4%

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

      3. +-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in a around 0 52.0%

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

      1. expm1-log1p-u 36.9%

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

      2. expm1-udef 36.6%

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

      3. quot-tan 36.6%

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

   6. Applied egg-rr 36.6%

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

      1. expm1-def 37.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

      2. expm1-log1p 52.2%

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

   8. Simplified 52.2%

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

   if -1.3999999999999999e-4 < b < 1.06e-4

   1. Initial program 98.5%

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

      1. associate-*r/ 98.4%

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

      2. *-commutative 98.4%

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

      3. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 98.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000106\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000102\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00014) (not (<= b 0.000102)))
   (* r (tan b))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 0.000102)) {
		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 <= (-0.00014d0)) .or. (.not. (b <= 0.000102d0))) 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 <= -0.00014) || !(b <= 0.000102)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00014) or not (b <= 0.000102):
		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 <= -0.00014) || !(b <= 0.000102))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00014) || ~((b <= 0.000102)))
		tmp = r * tan(b);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00014], N[Not[LessEqual[b, 0.000102]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999e-4 or 1.01999999999999999e-4 < b

   1. Initial program 51.4%

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

      1. associate-*r/ 51.4%

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

      2. *-commutative 51.4%

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

      3. +-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in a around 0 52.0%

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

      1. expm1-log1p-u 36.9%

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

      2. expm1-udef 36.6%

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

      3. quot-tan 36.6%

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

   6. Applied egg-rr 36.6%

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

      1. expm1-def 37.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

      2. expm1-log1p 52.2%

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

   8. Simplified 52.2%

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

   if -1.3999999999999999e-4 < b < 1.01999999999999999e-4

   1. Initial program 98.5%

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

      1. associate-*r/ 98.4%

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

      2. *-commutative 98.4%

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

      3. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 98.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000102\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 11: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

4. Taylor expanded in b around 0 53.9%

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

   1. mul-1-neg 53.9%

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

   2. unsub-neg 53.9%

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

6. Simplified 53.9%

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

7. Taylor expanded in a around 0 42.0%

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

8. Final simplification 42.0%

   \[\leadsto r \cdot \sin b \]

Alternative 12: 60.4% 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 75.5%

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

   1. associate-*r/ 75.5%

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

   2. *-commutative 75.5%

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

   3. +-commutative 75.5%

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

3. Simplified 75.5%

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

4. Taylor expanded in a around 0 61.6%

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

   1. expm1-log1p-u 54.2%

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

   2. expm1-udef 33.0%

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

   3. quot-tan 33.0%

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

6. Applied egg-rr 33.0%

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

   1. expm1-def 54.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

   2. expm1-log1p 61.7%

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

8. Simplified 61.7%

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

9. Final simplification 61.7%

   \[\leadsto r \cdot \tan b \]

Alternative 13: 35.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.5%

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

   2. *-commutative 75.5%

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

   3. +-commutative 75.5%

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

3. Simplified 75.5%

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

4. Taylor expanded in a around 0 61.6%

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

   1. clear-num 61.6%

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

   2. inv-pow 61.6%

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

6. Applied egg-rr 61.6%

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

   1. unpow-1 61.6%

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

8. Simplified 61.6%

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

9. Taylor expanded in b around 0 38.8%

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

10. Final simplification 38.8%

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

Alternative 14: 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 75.5%

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

   1. associate-/l* 75.4%

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

   2. +-commutative 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in a around 0 59.3%

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

   1. mul-1-neg 59.3%

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

   2. unsub-neg 59.3%

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

6. Simplified 59.3%

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

7. Taylor expanded in b around 0 37.9%

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

8. Final simplification 37.9%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023257 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))