rsin B (should all be same)

Percentage Accurate: 75.9% → 99.5%

Time: 13.4s

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: 75.9% 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, 0.4× 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 78.8%

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

   1. +-commutative 78.8%

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

3. Simplified 78.8%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;t\_0 \leq 10^{-17}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left|\tan b\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))))
   (if (<= t_0 -0.002)
     (* r (tan b))
     (if (<= t_0 1e-17) (* r (/ b (cos a))) (* r (fabs (tan b)))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double tmp;
	if (t_0 <= -0.002) {
		tmp = r * tan(b);
	} else if (t_0 <= 1e-17) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r * fabs(tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((b + a))
    if (t_0 <= (-0.002d0)) then
        tmp = r * tan(b)
    else if (t_0 <= 1d-17) then
        tmp = r * (b / cos(a))
    else
        tmp = r * abs(tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) / Math.cos((b + a));
	double tmp;
	if (t_0 <= -0.002) {
		tmp = r * Math.tan(b);
	} else if (t_0 <= 1e-17) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * Math.abs(Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	tmp = 0
	if t_0 <= -0.002:
		tmp = r * math.tan(b)
	elif t_0 <= 1e-17:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * math.fabs(math.tan(b))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = Float64(r * tan(b));
	elseif (t_0 <= 1e-17)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * abs(tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((b + a));
	tmp = 0.0;
	if (t_0 <= -0.002)
		tmp = r * tan(b);
	elseif (t_0 <= 1e-17)
		tmp = r * (b / cos(a));
	else
		tmp = r * abs(tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-17], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Abs[N[Tan[b], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -2e-3

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

   3. Simplified 63.3%

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

      1. add-log-exp 62.7%

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

   6. Applied egg-rr 62.7%

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

   7. Taylor expanded in a around 0 62.8%

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

      1. rem-log-exp 63.4%

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

      2. clear-num 63.2%

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

      3. un-div-inv 63.3%

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

      4. clear-num 63.3%

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

      5. quot-tan 63.4%

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

   9. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 63.5%

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

       2. /-rgt-identity 63.5%

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

   11. Simplified 63.5%

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

   if -2e-3 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 1.00000000000000007e-17

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   if 1.00000000000000007e-17 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

   3. Simplified 53.2%

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

      1. add-log-exp 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in a around 0 50.8%

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

      1. rem-log-exp 52.7%

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

      2. add-sqr-sqrt 52.0%

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

      3. sqrt-unprod 53.2%

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

      4. pow2 53.2%

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

      5. quot-tan 53.2%

         \[\leadsto r \cdot \sqrt{{\color{blue}{\tan b}}^{2}} \]

   9. Applied egg-rr 53.2%

      \[\leadsto r \cdot \color{blue}{\sqrt{{\tan b}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 53.2%

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

       2. rem-sqrt-square 53.2%

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

   11. Simplified 53.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.002:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 10^{-17}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left|\tan b\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ \mathbf{if}\;t\_0 \leq -0.002 \lor \neg \left(t\_0 \leq 10^{-17}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))))
   (if (or (<= t_0 -0.002) (not (<= t_0 1e-17)))
     (* r (tan b))
     (* r (/ b (cos a))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 1e-17)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((b + a))
    if ((t_0 <= (-0.002d0)) .or. (.not. (t_0 <= 1d-17))) 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 t_0 = Math.sin(b) / Math.cos((b + a));
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 1e-17)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	tmp = 0
	if (t_0 <= -0.002) or not (t_0 <= 1e-17):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	tmp = 0.0
	if ((t_0 <= -0.002) || !(t_0 <= 1e-17))
		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)
	t_0 = sin(b) / cos((b + a));
	tmp = 0.0;
	if ((t_0 <= -0.002) || ~((t_0 <= 1e-17)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.002], N[Not[LessEqual[t$95$0, 1e-17]], $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 (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -2e-3 or 1.00000000000000007e-17 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. add-log-exp 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in a around 0 57.1%

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

      1. rem-log-exp 58.3%

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

      2. clear-num 58.2%

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

      3. un-div-inv 58.2%

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

      4. clear-num 58.2%

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

      5. quot-tan 58.2%

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

   9. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 58.4%

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

       2. /-rgt-identity 58.4%

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

   11. Simplified 58.4%

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

   if -2e-3 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 1.00000000000000007e-17

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.002 \lor \neg \left(\frac{\sin b}{\cos \left(b + a\right)} \leq 10^{-17}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ \mathbf{if}\;t\_0 \leq -0.002 \lor \neg \left(t\_0 \leq 10^{-17}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))))
   (if (or (<= t_0 -0.002) (not (<= t_0 1e-17)))
     (* r (tan b))
     (* b (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 1e-17)) {
		tmp = r * tan(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) :: t_0
    real(8) :: tmp
    t_0 = sin(b) / cos((b + a))
    if ((t_0 <= (-0.002d0)) .or. (.not. (t_0 <= 1d-17))) then
        tmp = r * tan(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 t_0 = Math.sin(b) / Math.cos((b + a));
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 1e-17)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	tmp = 0
	if (t_0 <= -0.002) or not (t_0 <= 1e-17):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	tmp = 0.0
	if ((t_0 <= -0.002) || !(t_0 <= 1e-17))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((b + a));
	tmp = 0.0;
	if ((t_0 <= -0.002) || ~((t_0 <= 1e-17)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.002], N[Not[LessEqual[t$95$0, 1e-17]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -2e-3 or 1.00000000000000007e-17 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. add-log-exp 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in a around 0 57.1%

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

      1. rem-log-exp 58.3%

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

      2. clear-num 58.2%

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

      3. un-div-inv 58.2%

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

      4. clear-num 58.2%

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

      5. quot-tan 58.2%

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

   9. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 58.4%

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

       2. /-rgt-identity 58.4%

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

   11. Simplified 58.4%

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

   if -2e-3 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 1.00000000000000007e-17

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 99.7%

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

      1. associate-/l* 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -0.002 \lor \neg \left(\frac{\sin b}{\cos \left(b + a\right)} \leq 10^{-17}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \sin a \cdot 0} \end{array} \]

FPCore

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

C

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

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(a) * 0.0d0)))
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(a) * 0.0)));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / ((math.cos(b) * math.cos(a)) + (math.sin(a) * 0.0)))

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / ((cos(b) * cos(a)) + (sin(a) * 0.0)));
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[a], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

   1. +-commutative 78.8%

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

3. Simplified 78.8%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

   1. expm1-log1p-u 99.6%

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

   2. expm1-undefine 99.6%

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

8. Applied egg-rr 99.6%

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

   1. expm1-define 99.6%

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

10. Simplified 99.6%

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

    1. expm1-undefine 99.6%

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

    2. log1p-undefine 99.5%

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

    3. rem-exp-log 99.6%

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

    4. +-commutative 99.6%

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

12. Applied egg-rr 99.6%

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

13. Taylor expanded in b around 0 80.1%

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

14. Final simplification 80.1%

    \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a + \sin a \cdot 0} \]
15. Add Preprocessing

Alternative 6: 75.9% 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 78.8%

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

3. Final simplification 78.8%

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

Alternative 7: 59.6% 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 78.8%

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

   1. +-commutative 78.8%

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

3. Simplified 78.8%

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

   1. add-log-exp 41.0%

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

6. Applied egg-rr 41.0%

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

7. Taylor expanded in a around 0 40.9%

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

   1. rem-log-exp 58.9%

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

   2. clear-num 58.8%

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

   3. un-div-inv 58.7%

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

   4. clear-num 58.7%

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

   5. quot-tan 58.7%

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

9. Applied egg-rr 58.7%

   \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
10. Step-by-step derivation

    1. associate-/r/ 58.9%

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

    2. /-rgt-identity 58.9%

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

11. Simplified 58.9%

    \[\leadsto \color{blue}{r \cdot \tan b} \]
12. Add Preprocessing

Alternative 8: 34.1% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. +-commutative 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in b around 0 52.0%

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

   1. associate-/l* 52.0%

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

7. Simplified 52.0%

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

8. Taylor expanded in a around 0 32.2%

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

9. Final simplification 32.2%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))