rsin B (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 12.0s

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: 76.3% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. sin-neg N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-sin.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 2: 76.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. difference-of-squares N/A

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

   10. lift-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower--.f64 48.7

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

4. Applied rewrites 48.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

6. Applied rewrites 39.9%

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

8. Applied rewrites 76.3%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))