rsin A (should all be same)

Percentage Accurate: 76.7% → 99.5%

Time: 25.0s

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: 76.7% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. +-commutative 80.4%

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

3. Simplified 80.4%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. distribute-rgt-neg-in 99.5%

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

   5. *-commutative 99.5%

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

   6. fma-def 99.6%

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

   7. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. +-commutative 80.4%

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

3. Simplified 80.4%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. +-commutative 80.4%

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

3. Simplified 80.4%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in r around 0 99.5%

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

   1. associate-/l* 99.5%

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

   2. +-commutative 99.5%

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

   3. mul-1-neg 99.5%

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

   4. *-commutative 99.5%

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

   5. sub-neg 99.5%

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

   6. *-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (sin(b) * r) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))
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(b) * Math.sin(a)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. +-commutative 80.4%

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

3. Simplified 80.4%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0001 \lor \neg \left(b \leq 0.086\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0001) (not (<= b 0.086)))
   (* (sin b) (/ r (cos b)))
   (/ (* b r) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0001) || !(b <= 0.086)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (b * r) / cos((b + a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.0001d0)) .or. (.not. (b <= 0.086d0))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (b * r) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0001) || !(b <= 0.086)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (b * r) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0001) or not (b <= 0.086):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (b * r) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0001) || !(b <= 0.086))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0001) || ~((b <= 0.086)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (b * r) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0001], N[Not[LessEqual[b, 0.086]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.00000000000000005e-4 or 0.085999999999999993 < b

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-/l* 61.3%

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

      3. +-commutative 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in a around 0 61.2%

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

      1. associate-/l* 61.3%

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

      2. associate-/r/ 61.3%

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

   6. Simplified 61.3%

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

   if -1.00000000000000005e-4 < b < 0.085999999999999993

   1. Initial program 98.3%

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

      1. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in b around 0 98.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0001 \lor \neg \left(b \leq 0.086\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -22000000 \lor \neg \left(a \leq 0.0023\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -22000000.0) (not (<= a 0.0023)))
   (/ r (/ (cos a) (sin b)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -22000000.0) || !(a <= 0.0023)) {
		tmp = r / (cos(a) / sin(b));
	} else {
		tmp = sin(b) * (r / 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 <= (-22000000.0d0)) .or. (.not. (a <= 0.0023d0))) then
        tmp = r / (cos(a) / sin(b))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -22000000.0) || !(a <= 0.0023)) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -22000000.0) or not (a <= 0.0023):
		tmp = r / (math.cos(a) / math.sin(b))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -22000000.0) || !(a <= 0.0023))
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -22000000.0) || ~((a <= 0.0023)))
		tmp = r / (cos(a) / sin(b));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -22000000.0], N[Not[LessEqual[a, 0.0023]], $MachinePrecision]], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2e7 or 0.0023 < a

   1. Initial program 63.7%

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

      1. +-commutative 63.7%

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

   3. Simplified 63.7%

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

      1. log1p-expm1-u 48.2%

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

      2. log1p-udef 15.3%

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

      3. *-commutative 15.3%

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

   5. Applied egg-rr 15.3%

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

   6. Taylor expanded in b around 0 15.5%

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

   7. Taylor expanded in b around inf 64.1%

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

      1. associate-/l* 64.1%

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

   9. Simplified 64.1%

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

   if -2.2e7 < a < 0.0023

   1. Initial program 95.9%

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

      1. *-commutative 95.9%

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

      2. associate-/l* 95.8%

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

      3. +-commutative 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around 0 95.8%

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

      1. associate-/l* 95.7%

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

      2. associate-/r/ 95.9%

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

   6. Simplified 95.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -22000000 \lor \neg \left(a \leq 0.0023\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 7: 76.7% 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 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-/l* 80.3%

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

   3. +-commutative 80.3%

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

3. Simplified 80.3%

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

   1. clear-num 79.7%

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

   2. associate-/r/ 80.2%

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

   3. clear-num 80.4%

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

5. Applied egg-rr 80.4%

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

6. Final simplification 80.4%

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

Alternative 8: 76.7% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 80.4%

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

2. Final simplification 80.4%

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

Alternative 9: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -980000 \lor \neg \left(b \leq 105\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -980000.0) (not (<= b 105.0)))
   (* (sin b) r)
   (/ (* b r) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -980000.0) || !(b <= 105.0)) {
		tmp = sin(b) * r;
	} else {
		tmp = (b * r) / cos((b + a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-980000.0d0)) .or. (.not. (b <= 105.0d0))) then
        tmp = sin(b) * r
    else
        tmp = (b * r) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -980000.0) || !(b <= 105.0)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = (b * r) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -980000.0) or not (b <= 105.0):
		tmp = math.sin(b) * r
	else:
		tmp = (b * r) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -980000.0) || !(b <= 105.0))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -980000.0) || ~((b <= 105.0)))
		tmp = sin(b) * r;
	else
		tmp = (b * r) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -980000.0], N[Not[LessEqual[b, 105.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.8e5 or 105 < b

   1. Initial program 61.1%

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

      1. +-commutative 61.1%

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

   3. Simplified 61.1%

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

      1. log1p-expm1-u 34.8%

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

      2. log1p-udef 4.9%

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

      3. *-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in b around 0 4.5%

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

   7. Taylor expanded in a around 0 13.0%

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

   if -9.8e5 < b < 105

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in b around 0 96.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -980000 \lor \neg \left(b \leq 105\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 10: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -105 \lor \neg \left(b \leq 1.86\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -105.0) (not (<= b 1.86))) (* (sin b) r) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -105.0) || !(b <= 1.86)) {
		tmp = sin(b) * r;
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-105.0d0)) .or. (.not. (b <= 1.86d0))) then
        tmp = sin(b) * r
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -105.0) || !(b <= 1.86)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -105.0) or not (b <= 1.86):
		tmp = math.sin(b) * r
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -105.0) || !(b <= 1.86))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -105.0) || ~((b <= 1.86)))
		tmp = sin(b) * r;
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -105.0], N[Not[LessEqual[b, 1.86]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -105 or 1.8600000000000001 < b

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. log1p-expm1-u 35.4%

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

      2. log1p-udef 4.9%

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

      3. *-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in b around 0 4.4%

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

   7. Taylor expanded in a around 0 12.9%

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

   if -105 < b < 1.8600000000000001

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/l* 97.5%

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

      3. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in b around 0 97.2%

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

      1. associate-/l* 97.0%

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

      2. associate-/r/ 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -105 \lor \neg \left(b \leq 1.86\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -118 \lor \neg \left(b \leq 0.74\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -118.0) (not (<= b 0.74))) (* (sin b) r) (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -118.0) || !(b <= 0.74)) {
		tmp = sin(b) * r;
	} else {
		tmp = (b * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-118.0d0)) .or. (.not. (b <= 0.74d0))) then
        tmp = sin(b) * r
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -118.0) || !(b <= 0.74)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -118.0) or not (b <= 0.74):
		tmp = math.sin(b) * r
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -118.0) || !(b <= 0.74))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -118.0) || ~((b <= 0.74)))
		tmp = sin(b) * r;
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -118.0], N[Not[LessEqual[b, 0.74]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -118 or 0.73999999999999999 < b

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. log1p-expm1-u 35.4%

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

      2. log1p-udef 4.9%

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

      3. *-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in b around 0 4.4%

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

   7. Taylor expanded in a around 0 12.9%

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

   if -118 < b < 0.73999999999999999

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/l* 97.5%

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

      3. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in b around 0 97.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -118 \lor \neg \left(b \leq 0.74\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 12: 38.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. +-commutative 80.4%

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

3. Simplified 80.4%

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

   1. log1p-expm1-u 55.3%

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

   2. log1p-udef 16.3%

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

   3. *-commutative 16.3%

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

5. Applied egg-rr 16.3%

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

6. Taylor expanded in b around 0 16.1%

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

7. Taylor expanded in a around 0 36.0%

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

8. Final simplification 36.0%

   \[\leadsto \sin b \cdot r \]

Alternative 13: 34.8% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-/l* 80.3%

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

   3. +-commutative 80.3%

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

3. Simplified 80.3%

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

4. Taylor expanded in b around 0 52.6%

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

   1. associate-/l* 52.4%

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

   2. associate-/r/ 52.5%

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

6. Simplified 52.5%

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

7. Taylor expanded in a around 0 31.6%

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

8. Final simplification 31.6%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023287 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))