rsin A (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 21.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. prod-diff 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-define 99.5%

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

   4. associate-+l+ 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-rgt-neg-in 99.5%

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

   7. *-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

   1. fma-define 99.5%

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

   2. fma-define 99.5%

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

   3. sin-neg 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (- (* (cos b) (cos a)) (log1p (expm1 (* (sin b) (sin a))))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / ((cos(b) * cos(a)) - log1p(expm1((sin(b) * sin(a))))));
}

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / ((Math.cos(b) * Math.cos(a)) - Math.log1p(Math.expm1((Math.sin(b) * Math.sin(a))))));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / ((math.cos(b) * math.cos(a)) - math.log1p(math.expm1((math.sin(b) * math.sin(a))))))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / Float64(Float64(cos(b) * cos(a)) - log1p(expm1(Float64(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[Log[1 + N[(Exp[N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. log1p-expm1-u 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)} \]
10. Add Preprocessing

Alternative 3: 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 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 78.0% accurate, 0.7× speedup

Math

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

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

Derivation

1. Initial program 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. sin-mult 76.7%

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

   3. +-commutative 76.7%

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

   4. div-sub 76.7%

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

   5. cos-sum 77.4%

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

   6. *-commutative 77.4%

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

   7. cancel-sign-sub-inv 77.4%

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

   8. add-sqr-sqrt 37.6%

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

   9. sqrt-unprod 76.6%

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

   10. sqr-neg 76.6%

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

   11. sqrt-unprod 39.0%

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

   12. add-sqr-sqrt 75.9%

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

   13. *-commutative 75.9%

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

   14. cos-diff 76.5%

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

8. Applied egg-rr 76.5%

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

   1. +-inverses 76.5%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]

10. Simplified 76.5%

    \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]

11. Final simplification 76.5%

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

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00034 \lor \neg \left(a \leq 9.8 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00034) (not (<= a 9.8e-5)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00034) || !(a <= 9.8e-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 <= (-0.00034d0)) .or. (.not. (a <= 9.8d-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 <= -0.00034) || !(a <= 9.8e-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 <= -0.00034) or not (a <= 9.8e-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 <= -0.00034) || !(a <= 9.8e-5))
		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 <= -0.00034) || ~((a <= 9.8e-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, -0.00034], N[Not[LessEqual[a, 9.8e-5]], $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 < -3.4e-4 or 9.8e-5 < a

   1. Initial program 53.1%

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

      1. associate-/l* 53.2%

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

      2. remove-double-neg 53.2%

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

      3. remove-double-neg 53.2%

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

      4. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in b around 0 52.4%

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

   if -3.4e-4 < a < 9.8e-5

   1. Initial program 99.2%

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

      1. associate-/l* 99.3%

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

      2. remove-double-neg 99.3%

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

      3. remove-double-neg 99.3%

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

      4. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00034 \lor \neg \left(a \leq 9.8 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00014:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.00014:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.00014)
   (* (sin b) (/ r (cos a)))
   (if (<= a 0.00014) (* r (/ (sin b) (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00014) {
		tmp = sin(b) * (r / cos(a));
	} else if (a <= 0.00014) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = r * (sin(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 (a <= (-0.00014d0)) then
        tmp = sin(b) * (r / cos(a))
    else if (a <= 0.00014d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00014) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else if (a <= 0.00014) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.00014:
		tmp = math.sin(b) * (r / math.cos(a))
	elif a <= 0.00014:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.00014)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	elseif (a <= 0.00014)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.00014)
		tmp = sin(b) * (r / cos(a));
	elseif (a <= 0.00014)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.00014], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00014], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999e-4

   1. Initial program 51.1%

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

      1. +-commutative 51.1%

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

   3. Simplified 51.1%

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

      1. add-log-exp 51.1%

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

   6. Applied egg-rr 51.1%

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

   7. Taylor expanded in b around 0 50.0%

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

      1. rem-log-exp 50.1%

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

      2. log1p-expm1-u 50.0%

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

   9. Applied egg-rr 50.0%

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

   10. Taylor expanded in r around 0 50.1%

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

       1. *-commutative 50.1%

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

       2. associate-*r/ 50.1%

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

   12. Simplified 50.1%

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

   if -1.3999999999999999e-4 < a < 1.3999999999999999e-4

   1. Initial program 99.2%

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

      1. associate-/l* 99.3%

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

      2. remove-double-neg 99.3%

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

      3. remove-double-neg 99.3%

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

      4. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.0%

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

   if 1.3999999999999999e-4 < a

   1. Initial program 55.2%

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

      1. associate-/l* 55.3%

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

      2. remove-double-neg 55.3%

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

      3. remove-double-neg 55.3%

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

      4. +-commutative 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in b around 0 54.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00014:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.00014:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% 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 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

5. Final simplification 76.1%

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

Alternative 8: 55.6% 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 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

5. Taylor expanded in b around 0 52.0%

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

6. Final simplification 52.0%

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

Alternative 9: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \lor \neg \left(b \leq 42000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.32) (not (<= b 42000000.0)))
   (* r (sin b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.32) || !(b <= 42000000.0)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.32d0)) .or. (.not. (b <= 42000000.0d0))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.32) || !(b <= 42000000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.32) or not (b <= 42000000.0):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.32) || !(b <= 42000000.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.32) || ~((b <= 42000000.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.32], N[Not[LessEqual[b, 42000000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.32000000000000006 or 4.2e7 < b

   1. Initial program 55.8%

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

      1. +-commutative 55.8%

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

   3. Simplified 55.8%

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

      1. add-log-exp 55.4%

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

   6. Applied egg-rr 55.4%

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

   7. Taylor expanded in b around 0 11.2%

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

   8. Taylor expanded in a around 0 11.3%

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

   if -1.32000000000000006 < b < 4.2e7

   1. Initial program 98.9%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. remove-double-neg 99.0%

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

      4. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in b around 0 98.3%

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

      1. associate-/l* 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \lor \neg \left(b \leq 42000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. +-commutative 76.0%

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

3. Simplified 76.0%

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

   1. add-log-exp 75.7%

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

6. Applied egg-rr 75.7%

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

7. Taylor expanded in b around 0 51.8%

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

8. Taylor expanded in a around 0 37.7%

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

9. Final simplification 37.7%

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

Alternative 11: 35.3% 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 76.0%

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

   1. associate-/l* 76.1%

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

   2. remove-double-neg 76.1%

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

   3. remove-double-neg 76.1%

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

   4. +-commutative 76.1%

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

3. Simplified 76.1%

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

5. Taylor expanded in b around 0 48.1%

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

   1. associate-/l* 48.1%

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

7. Simplified 48.1%

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

8. Taylor expanded in a around 0 33.9%

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

9. Final simplification 33.9%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))