rsin B (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 13.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around inf 99.5%

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

   1. sub-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-out 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-def 99.6%

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

   6. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+21}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.9:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -5e+21)
   (* r (tan b))
   (if (<= b 1.9) (* r (/ b (cos (+ b a)))) (/ (sin b) (/ (cos b) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -5e+21) {
		tmp = r * tan(b);
	} else if (b <= 1.9) {
		tmp = r * (b / cos((b + a)));
	} else {
		tmp = sin(b) / (cos(b) / r);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-5d+21)) then
        tmp = r * tan(b)
    else if (b <= 1.9d0) then
        tmp = r * (b / cos((b + a)))
    else
        tmp = sin(b) / (cos(b) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -5e+21) {
		tmp = r * Math.tan(b);
	} else if (b <= 1.9) {
		tmp = r * (b / Math.cos((b + a)));
	} else {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -5e+21:
		tmp = r * math.tan(b)
	elif b <= 1.9:
		tmp = r * (b / math.cos((b + a)))
	else:
		tmp = math.sin(b) / (math.cos(b) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -5e+21)
		tmp = Float64(r * tan(b));
	elseif (b <= 1.9)
		tmp = Float64(r * Float64(b / cos(Float64(b + a))));
	else
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -5e+21)
		tmp = r * tan(b);
	elseif (b <= 1.9)
		tmp = r * (b / cos((b + a)));
	else
		tmp = sin(b) / (cos(b) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -5e+21], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5e21

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

   3. Simplified 59.9%

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

      1. associate-*r/ 59.9%

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

      2. expm1-log1p-u 42.2%

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

      3. *-commutative 42.2%

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

   5. Applied egg-rr 42.2%

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

   6. Taylor expanded in a around 0 41.8%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b \cdot r}{\color{blue}{\cos b}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 60.0%

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

      2. div-inv 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-*l* 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. expm1-log1p-u 47.6%

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

      2. expm1-udef 47.4%

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

      3. un-div-inv 47.4%

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

      4. quot-tan 47.4%

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

   10. Applied egg-rr 47.4%

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

       1. expm1-def 47.7%

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

       2. expm1-log1p 60.1%

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

   12. Simplified 60.1%

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

   if -5e21 < b < 1.8999999999999999

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in b around 0 96.6%

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

   if 1.8999999999999999 < b

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-/r/ 49.6%

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

      3. +-commutative 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in a around 0 50.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+21}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.9:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \]

Alternative 4: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

2. Final simplification 75.3%

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

Alternative 5: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+21} \lor \neg \left(b \leq 1.9\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5e+21) (not (<= b 1.9)))
   (* r (tan b))
   (* r (/ b (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5e+21) || !(b <= 1.9)) {
		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 <= (-5d+21)) .or. (.not. (b <= 1.9d0))) 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 <= -5e+21) || !(b <= 1.9)) {
		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 <= -5e+21) or not (b <= 1.9):
		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 <= -5e+21) || !(b <= 1.9))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -5e+21) || ~((b <= 1.9)))
		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, -5e+21], N[Not[LessEqual[b, 1.9]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5e21 or 1.8999999999999999 < b

   1. Initial program 54.9%

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

      1. +-commutative 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r/ 54.9%

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

      2. expm1-log1p-u 39.5%

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

      3. *-commutative 39.5%

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

   5. Applied egg-rr 39.5%

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

   6. Taylor expanded in a around 0 39.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b \cdot r}{\color{blue}{\cos b}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 55.5%

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

      2. div-inv 55.5%

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

      3. *-commutative 55.5%

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

      4. associate-*l* 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. expm1-log1p-u 44.0%

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

      2. expm1-udef 43.8%

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

      3. un-div-inv 43.9%

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

      4. quot-tan 43.9%

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

   10. Applied egg-rr 43.9%

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

       1. expm1-def 44.1%

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

       2. expm1-log1p 55.6%

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

   12. Simplified 55.6%

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

   if -5e21 < b < 1.8999999999999999

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in b around 0 96.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+21} \lor \neg \left(b \leq 1.9\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 6: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+21} \lor \neg \left(b \leq 1.9\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5e+21) (not (<= b 1.9))) (* r (tan b)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5e+21) || !(b <= 1.9)) {
		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 <= (-5d+21)) .or. (.not. (b <= 1.9d0))) 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 <= -5e+21) || !(b <= 1.9)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -5e+21) or not (b <= 1.9):
		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 <= -5e+21) || !(b <= 1.9))
		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 <= -5e+21) || ~((b <= 1.9)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -5e+21], N[Not[LessEqual[b, 1.9]], $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 < -5e21 or 1.8999999999999999 < b

   1. Initial program 54.9%

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

      1. +-commutative 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r/ 54.9%

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

      2. expm1-log1p-u 39.5%

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

      3. *-commutative 39.5%

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

   5. Applied egg-rr 39.5%

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

   6. Taylor expanded in a around 0 39.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b \cdot r}{\color{blue}{\cos b}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 55.5%

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

      2. div-inv 55.5%

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

      3. *-commutative 55.5%

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

      4. associate-*l* 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. expm1-log1p-u 44.0%

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

      2. expm1-udef 43.8%

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

      3. un-div-inv 43.9%

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

      4. quot-tan 43.9%

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

   10. Applied egg-rr 43.9%

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

       1. expm1-def 44.1%

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

       2. expm1-log1p 55.6%

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

   12. Simplified 55.6%

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

   if -5e21 < b < 1.8999999999999999

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in b around 0 96.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+21} \lor \neg \left(b \leq 1.9\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 7: 59.9% 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 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

   1. associate-*r/ 75.2%

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

   2. expm1-log1p-u 59.1%

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

   3. *-commutative 59.1%

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

5. Applied egg-rr 59.1%

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

6. Taylor expanded in a around 0 46.4%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b \cdot r}{\color{blue}{\cos b}}\right)\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 59.1%

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

   2. div-inv 59.1%

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

   3. *-commutative 59.1%

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

   4. associate-*l* 59.1%

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

8. Applied egg-rr 59.1%

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

   1. expm1-log1p-u 53.3%

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

   2. expm1-udef 37.0%

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

   3. un-div-inv 37.0%

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

   4. quot-tan 37.0%

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

10. Applied egg-rr 37.0%

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

    1. expm1-def 53.3%

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

    2. expm1-log1p 59.2%

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

12. Simplified 59.2%

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

13. Final simplification 59.2%

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

Alternative 8: 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 75.3%

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

   1. +-commutative 75.3%

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

3. Simplified 75.3%

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

4. Taylor expanded in b around 0 48.9%

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

5. Taylor expanded in a around 0 32.6%

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

6. Final simplification 32.6%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023182 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))