rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 11.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -6.4e-6)
   (* (/ (sin b) (cos a)) r)
   (if (<= a 4.4e-9) (* (tan b) r) (/ (* (sin b) r) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -6.4e-6) {
		tmp = (sin(b) / cos(a)) * r;
	} else if (a <= 4.4e-9) {
		tmp = tan(b) * r;
	} else {
		tmp = (sin(b) * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.4d-6)) then
        tmp = (sin(b) / cos(a)) * r
    else if (a <= 4.4d-9) then
        tmp = tan(b) * r
    else
        tmp = (sin(b) * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -6.4e-6) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else if (a <= 4.4e-9) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -6.4e-6:
		tmp = (math.sin(b) / math.cos(a)) * r
	elif a <= 4.4e-9:
		tmp = math.tan(b) * r
	else:
		tmp = (math.sin(b) * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -6.4e-6)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	elseif (a <= 4.4e-9)
		tmp = Float64(tan(b) * r);
	else
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -6.4e-6)
		tmp = (sin(b) / cos(a)) * r;
	elseif (a <= 4.4e-9)
		tmp = tan(b) * r;
	else
		tmp = (sin(b) * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -6.4e-6], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[a, 4.4e-9], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.3999999999999997e-6

   1. Initial program 61.5%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 61.2

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

   7. Applied rewrites 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 61.2

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

   9. Applied rewrites 61.2%

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

   if -6.3999999999999997e-6 < a < 4.3999999999999997e-9

   1. Initial program 99.5%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.4

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.5%

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

   if 4.3999999999999997e-9 < a

   1. Initial program 48.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 48.8

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

   5. Applied rewrites 48.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -6.4e-6)
   (* (/ (sin b) (cos a)) r)
   (if (<= a 4.4e-9) (* (tan b) r) (* (/ r (cos a)) (sin b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -6.4e-6) {
		tmp = (sin(b) / cos(a)) * r;
	} else if (a <= 4.4e-9) {
		tmp = tan(b) * r;
	} else {
		tmp = (r / cos(a)) * sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.4d-6)) then
        tmp = (sin(b) / cos(a)) * r
    else if (a <= 4.4d-9) then
        tmp = tan(b) * r
    else
        tmp = (r / cos(a)) * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -6.4e-6) {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	} else if (a <= 4.4e-9) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -6.4e-6:
		tmp = (math.sin(b) / math.cos(a)) * r
	elif a <= 4.4e-9:
		tmp = math.tan(b) * r
	else:
		tmp = (r / math.cos(a)) * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -6.4e-6)
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	elseif (a <= 4.4e-9)
		tmp = Float64(tan(b) * r);
	else
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -6.4e-6)
		tmp = (sin(b) / cos(a)) * r;
	elseif (a <= 4.4e-9)
		tmp = tan(b) * r;
	else
		tmp = (r / cos(a)) * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -6.4e-6], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[a, 4.4e-9], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.3999999999999997e-6

   1. Initial program 61.5%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 61.2

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

   7. Applied rewrites 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 61.2

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

   9. Applied rewrites 61.2%

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

   if -6.3999999999999997e-6 < a < 4.3999999999999997e-9

   1. Initial program 99.5%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.4

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.5%

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

   if 4.3999999999999997e-9 < a

   1. Initial program 48.7%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 48.8

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

   7. Applied rewrites 48.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.7

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

   9. Applied rewrites 48.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos a} \cdot \sin b\\ \mathbf{if}\;a \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos a)) (sin b))))
   (if (<= a -6.4e-6) t_0 (if (<= a 4.4e-9) (* (tan b) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(a)) * sin(b);
	double tmp;
	if (a <= -6.4e-6) {
		tmp = t_0;
	} else if (a <= 4.4e-9) {
		tmp = tan(b) * r;
	} else {
		tmp = t_0;
	}
	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 = (r / cos(a)) * sin(b)
    if (a <= (-6.4d-6)) then
        tmp = t_0
    else if (a <= 4.4d-9) then
        tmp = tan(b) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r / Math.cos(a)) * Math.sin(b);
	double tmp;
	if (a <= -6.4e-6) {
		tmp = t_0;
	} else if (a <= 4.4e-9) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(a)) * math.sin(b)
	tmp = 0
	if a <= -6.4e-6:
		tmp = t_0
	elif a <= 4.4e-9:
		tmp = math.tan(b) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(a)) * sin(b))
	tmp = 0.0
	if (a <= -6.4e-6)
		tmp = t_0;
	elseif (a <= 4.4e-9)
		tmp = Float64(tan(b) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(a)) * sin(b);
	tmp = 0.0;
	if (a <= -6.4e-6)
		tmp = t_0;
	elseif (a <= 4.4e-9)
		tmp = tan(b) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.4e-6], t$95$0, If[LessEqual[a, 4.4e-9], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.3999999999999997e-6 or 4.3999999999999997e-9 < a

   1. Initial program 55.9%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 55.8

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

   7. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.7

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

   9. Applied rewrites 55.7%

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

   if -6.3999999999999997e-6 < a < 4.3999999999999997e-9

   1. Initial program 99.5%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.4

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Final simplification 79.2%

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

Alternative 7: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 79.2

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

4. Applied rewrites 79.2%

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

Alternative 8: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -6.2e-6) t_0 (if (<= b 3.9e-10) (/ (* b r) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -6.2e-6) {
		tmp = t_0;
	} else if (b <= 3.9e-10) {
		tmp = (b * r) / cos(a);
	} else {
		tmp = t_0;
	}
	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 = tan(b) * r
    if (b <= (-6.2d-6)) then
        tmp = t_0
    else if (b <= 3.9d-10) then
        tmp = (b * r) / cos(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -6.2e-6) {
		tmp = t_0;
	} else if (b <= 3.9e-10) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -6.2e-6:
		tmp = t_0
	elif b <= 3.9e-10:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -6.2e-6)
		tmp = t_0;
	elseif (b <= 3.9e-10)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -6.2e-6)
		tmp = t_0;
	elseif (b <= 3.9e-10)
		tmp = (b * r) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -6.2e-6], t$95$0, If[LessEqual[b, 3.9e-10], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.1999999999999999e-6 or 3.9e-10 < b

   1. Initial program 61.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 60.4

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 60.5%

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

   if -6.1999999999999999e-6 < b < 3.9e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\tan b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -6.2e-6) t_0 (if (<= b 3.9e-10) (* (/ b (cos a)) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -6.2e-6) {
		tmp = t_0;
	} else if (b <= 3.9e-10) {
		tmp = (b / cos(a)) * r;
	} else {
		tmp = t_0;
	}
	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 = tan(b) * r
    if (b <= (-6.2d-6)) then
        tmp = t_0
    else if (b <= 3.9d-10) then
        tmp = (b / cos(a)) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -6.2e-6) {
		tmp = t_0;
	} else if (b <= 3.9e-10) {
		tmp = (b / Math.cos(a)) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -6.2e-6:
		tmp = t_0
	elif b <= 3.9e-10:
		tmp = (b / math.cos(a)) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -6.2e-6)
		tmp = t_0;
	elseif (b <= 3.9e-10)
		tmp = Float64(Float64(b / cos(a)) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -6.2e-6)
		tmp = t_0;
	elseif (b <= 3.9e-10)
		tmp = (b / cos(a)) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -6.2e-6], t$95$0, If[LessEqual[b, 3.9e-10], N[(N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.1999999999999999e-6 or 3.9e-10 < b

   1. Initial program 61.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 60.4

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 60.5%

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

   if -6.1999999999999999e-6 < b < 3.9e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{b}{\cos a} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\tan b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. inv-pow N/A

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

   5. lower-pow.f64 79.1

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

4. Applied rewrites 79.1%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cos.f64 65.4

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

7. Applied rewrites 65.4%

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

9. Applied rewrites 65.4%

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

10. Final simplification 65.4%

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

Alternative 11: 34.8% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 49.3

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

5. Applied rewrites 49.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))