rsin A (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 12.4s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 75.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 75.9

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

4. Applied egg-rr 75.9%

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

   1. cos-sum N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-neg N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-fma.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

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

Derivation

1. Initial program 75.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 75.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 75.9

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

4. Applied egg-rr 75.9%

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

   1. cos-sum N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-*.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 75.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 75.9

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

4. Applied egg-rr 75.9%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. lift--.f64 N/A

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

   4. difference-of-squares N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 75.9

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

6. Applied egg-rr 75.9%

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

7. Final simplification 75.9%

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

Alternative 4: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.7e-5)
   (* r (/ (sin b) (cos b)))
   (if (<= b 2.2e-36) (* b (/ r (cos a))) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.7e-5) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 2.2e-36) {
		tmp = b * (r / cos(a));
	} else {
		tmp = (r * sin(b)) / cos(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 (b <= (-2.7d-5)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 2.2d-36) then
        tmp = b * (r / cos(a))
    else
        tmp = (r * sin(b)) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.7e-5) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 2.2e-36) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.7e-5:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 2.2e-36:
		tmp = b * (r / math.cos(a))
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.7e-5)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 2.2e-36)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.7e-5)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 2.2e-36)
		tmp = b * (r / cos(a));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.7e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-36], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6999999999999999e-5

   1. Initial program 44.6%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 44.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 44.6

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 45.6

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

   7. Simplified 45.6%

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

   if -2.6999999999999999e-5 < b < 2.1999999999999999e-36

   1. Initial program 99.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 2.1999999999999999e-36 < b

   1. Initial program 63.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 64.3

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

   5. Simplified 64.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -2.7e-5) t_0 (if (<= b 2.2e-36) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(b));
	double tmp;
	if (b <= -2.7e-5) {
		tmp = t_0;
	} else if (b <= 2.2e-36) {
		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 = r * (sin(b) / cos(b))
    if (b <= (-2.7d-5)) then
        tmp = t_0
    else if (b <= 2.2d-36) 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 = r * (Math.sin(b) / Math.cos(b));
	double tmp;
	if (b <= -2.7e-5) {
		tmp = t_0;
	} else if (b <= 2.2e-36) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(b))
	tmp = 0
	if b <= -2.7e-5:
		tmp = t_0
	elif b <= 2.2e-36:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(b)))
	tmp = 0.0
	if (b <= -2.7e-5)
		tmp = t_0;
	elseif (b <= 2.2e-36)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(b));
	tmp = 0.0;
	if (b <= -2.7e-5)
		tmp = t_0;
	elseif (b <= 2.2e-36)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.7e-5], t$95$0, If[LessEqual[b, 2.2e-36], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6999999999999999e-5 or 2.1999999999999999e-36 < b

   1. Initial program 55.6%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 55.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 55.6

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 56.4

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

   7. Simplified 56.4%

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

   if -2.6999999999999999e-5 < b < 2.1999999999999999e-36

   1. Initial program 99.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.5

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos b)))))
   (if (<= b -2.5e-5) t_0 (if (<= b 2.2e-36) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(b));
	double tmp;
	if (b <= -2.5e-5) {
		tmp = t_0;
	} else if (b <= 2.2e-36) {
		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 = sin(b) * (r / cos(b))
    if (b <= (-2.5d-5)) then
        tmp = t_0
    else if (b <= 2.2d-36) 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.sin(b) * (r / Math.cos(b));
	double tmp;
	if (b <= -2.5e-5) {
		tmp = t_0;
	} else if (b <= 2.2e-36) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(b))
	tmp = 0
	if b <= -2.5e-5:
		tmp = t_0
	elif b <= 2.2e-36:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(b)))
	tmp = 0.0
	if (b <= -2.5e-5)
		tmp = t_0;
	elseif (b <= 2.2e-36)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(b));
	tmp = 0.0;
	if (b <= -2.5e-5)
		tmp = t_0;
	elseif (b <= 2.2e-36)
		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[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e-5], t$95$0, If[LessEqual[b, 2.2e-36], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.50000000000000012e-5 or 2.1999999999999999e-36 < b

   1. Initial program 55.6%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 55.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 55.6

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 56.4

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

   7. Simplified 56.4%

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

   if -2.50000000000000012e-5 < b < 2.1999999999999999e-36

   1. Initial program 99.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.5

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% 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 75.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 75.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 75.9

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

4. Applied egg-rr 75.9%

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

5. Final simplification 75.9%

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

Alternative 8: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 75.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 75.9

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

4. Applied egg-rr 75.9%

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

5. Final simplification 75.9%

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

Alternative 9: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 75.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 75.9

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

4. Applied egg-rr 75.9%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 55.2

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

7. Simplified 55.2%

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

8. Final simplification 55.2%

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

Alternative 10: 54.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{r}{\sin a}\\ \mathbf{if}\;b \leq -5000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55000000000:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (/ r (sin a)))))
   (if (<= b -5000000.0)
     t_0
     (if (<= b 55000000000.0)
       (* (/ r (cos (+ b a))) (fma b (* -0.16666666666666666 (* b b)) b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = -(r / sin(a));
	double tmp;
	if (b <= -5000000.0) {
		tmp = t_0;
	} else if (b <= 55000000000.0) {
		tmp = (r / cos((b + a))) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(-Float64(r / sin(a)))
	tmp = 0.0
	if (b <= -5000000.0)
		tmp = t_0;
	elseif (b <= 55000000000.0)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[(r / N[Sin[a], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[b, -5000000.0], t$95$0, If[LessEqual[b, 55000000000.0], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5e6 or 5.5e10 < b

   1. Initial program 51.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. frac-2neg N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. times-frac N/A

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

      12. unpow-prod-down N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 10.2

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

   7. Simplified 10.2%

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

   8. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 9.9

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

   10. Simplified 9.9%

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

   if -5e6 < b < 5.5e10

   1. Initial program 98.7%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 98.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot {b}^{2}}, b\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, \frac{-1}{6} \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

      7. lower-*.f64 95.3

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

   7. Simplified 95.3%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4600000000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{r}{\sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 4600000000.0) (* b (/ r (cos a))) (- (/ r (sin a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 4600000000.0) {
		tmp = b * (r / cos(a));
	} else {
		tmp = -(r / sin(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 <= 4600000000.0d0) then
        tmp = b * (r / cos(a))
    else
        tmp = -(r / sin(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 4600000000.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = -(r / Math.sin(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 4600000000.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = -(r / math.sin(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 4600000000.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(-Float64(r / sin(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 4600000000.0)
		tmp = b * (r / cos(a));
	else
		tmp = -(r / sin(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 4600000000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(r / N[Sin[a], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 4.6e9

   1. Initial program 82.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 68.9

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

   5. Simplified 68.9%

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

      1. *-commutative N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.9

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

   7. Applied egg-rr 68.9%

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

   if 4.6e9 < b

   1. Initial program 59.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. frac-2neg N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. times-frac N/A

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

      12. unpow-prod-down N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

   4. Applied egg-rr 58.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 9.0

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

   7. Simplified 9.0%

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

   8. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 8.9

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

   10. Simplified 8.9%

       \[\leadsto \color{blue}{-\frac{r}{\sin a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 36.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4200000000:\\ \;\;\;\;r \cdot b\\ \mathbf{else}:\\ \;\;\;\;-\frac{r}{\sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 4200000000.0) (* r b) (- (/ r (sin a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 4200000000.0) {
		tmp = r * b;
	} else {
		tmp = -(r / sin(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 <= 4200000000.0d0) then
        tmp = r * b
    else
        tmp = -(r / sin(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 4200000000.0) {
		tmp = r * b;
	} else {
		tmp = -(r / Math.sin(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 4200000000.0:
		tmp = r * b
	else:
		tmp = -(r / math.sin(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 4200000000.0)
		tmp = Float64(r * b);
	else
		tmp = Float64(-Float64(r / sin(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 4200000000.0)
		tmp = r * b;
	else
		tmp = -(r / sin(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 4200000000.0], N[(r * b), $MachinePrecision], (-N[(r / N[Sin[a], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 4.2e9

   1. Initial program 82.3%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 68.9

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

   5. Simplified 68.9%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.9

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

   8. Simplified 49.9%

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

   if 4.2e9 < b

   1. Initial program 59.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. frac-2neg N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. times-frac N/A

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

      12. unpow-prod-down N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

   4. Applied egg-rr 58.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 9.0

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

   7. Simplified 9.0%

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

   8. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 8.9

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

   10. Simplified 8.9%

       \[\leadsto \color{blue}{-\frac{r}{\sin a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.2% accurate, 36.7× 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 75.8%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 50.6

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

5. Simplified 50.6%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 36.8

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

8. Simplified 36.8%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))