rsin B (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 19.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. associate-*r/ 72.2%

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

   2. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-define 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. associate-*r/ 72.2%

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

   2. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 77.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. associate-*r/ 72.2%

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

   2. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-define 99.4%

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

6. Applied egg-rr 99.4%

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

   1. fma-undefine 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

8. Applied egg-rr 73.8%

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

9. Final simplification 73.8%

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

Alternative 5: 77.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. associate-*r/ 72.2%

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

   2. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-define 99.4%

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

6. Applied egg-rr 99.4%

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

   1. fma-undefine 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. add-log-exp 98.9%

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

   4. cos-sum 71.8%

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

   5. add-sqr-sqrt 71.6%

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

   6. sqrt-unprod 71.8%

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

   7. prod-exp 71.8%

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

   8. cos-sum 73.0%

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

   9. cancel-sign-sub-inv 73.0%

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

   10. add-sqr-sqrt 36.9%

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

   11. sqrt-unprod 72.1%

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

   12. sqr-neg 72.1%

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

   13. sqrt-unprod 35.2%

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

   14. add-sqr-sqrt 71.6%

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

8. Applied egg-rr 73.7%

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

9. Final simplification 73.7%

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

Alternative 6: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00021 \lor \neg \left(a \leq 0.00105\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00021) (not (<= a 0.00105)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00021) || !(a <= 0.00105)) {
		tmp = r * (sin(b) / 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 ((a <= (-0.00021d0)) .or. (.not. (a <= 0.00105d0))) then
        tmp = r * (sin(b) / 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 ((a <= -0.00021) || !(a <= 0.00105)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00021) or not (a <= 0.00105):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00021) || !(a <= 0.00105))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00021) || ~((a <= 0.00105)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00021], N[Not[LessEqual[a, 0.00105]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1000000000000001e-4 or 0.00104999999999999994 < a

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in b around 0 49.7%

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

   if -2.1000000000000001e-4 < a < 0.00104999999999999994

   1. Initial program 98.0%

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

      1. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in a around 0 98.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00021 \lor \neg \left(a \leq 0.00105\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0018 \lor \neg \left(a \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.0018) (not (<= a 9.2e-5)))
   (/ (* r (sin b)) (cos a))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0018) || !(a <= 9.2e-5)) {
		tmp = (r * sin(b)) / 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 ((a <= (-0.0018d0)) .or. (.not. (a <= 9.2d-5))) then
        tmp = (r * sin(b)) / 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 ((a <= -0.0018) || !(a <= 9.2e-5)) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.0018) or not (a <= 9.2e-5):
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.0018) || !(a <= 9.2e-5))
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.0018) || ~((a <= 9.2e-5)))
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.0018], N[Not[LessEqual[a, 9.2e-5]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0018 or 9.20000000000000001e-5 < a

   1. Initial program 49.7%

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

      1. associate-*r/ 49.8%

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

      2. +-commutative 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in b around 0 49.7%

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

   if -0.0018 < a < 9.20000000000000001e-5

   1. Initial program 98.0%

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

      1. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in a around 0 98.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0018 \lor \neg \left(a \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% 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 72.2%

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

3. Final simplification 72.2%

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

Alternative 9: 76.5% 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 72.2%

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

   1. associate-*r/ 72.2%

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

   2. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. *-commutative 72.2%

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

   2. associate-/l* 72.2%

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

6. Applied egg-rr 72.2%

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

7. Final simplification 72.2%

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

Alternative 10: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in b around 0 51.6%

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

6. Final simplification 51.6%

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

Alternative 11: 50.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in b around 0 47.6%

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

   1. associate-/l* 47.6%

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

7. Simplified 47.6%

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

8. Final simplification 47.6%

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

Alternative 12: 34.2% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in b around 0 47.6%

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

   1. associate-/l* 47.6%

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

7. Simplified 47.6%

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

8. Taylor expanded in a around 0 31.0%

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

   1. *-commutative 31.0%

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

10. Simplified 31.0%

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

11. Final simplification 31.0%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))