Trigonometry A

Percentage Accurate: 99.8% → 99.7%

Time: 9.2s

Alternatives: 13

Speedup: N/A×

Specification

\[0 \leq e \land e \leq 1\]

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(v + v\right), e \cdot \left(-e\right), 1\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  (* (sin v) (* e (- 1.0 (* e (cos v)))))
  (fma (+ 0.5 (* 0.5 (cos (+ v v)))) (* e (- e)) 1.0)))

C

double code(double e, double v) {
	return (sin(v) * (e * (1.0 - (e * cos(v))))) / fma((0.5 + (0.5 * cos((v + v)))), (e * -e), 1.0);
}

Julia

function code(e, v)
	return Float64(Float64(sin(v) * Float64(e * Float64(1.0 - Float64(e * cos(v))))) / fma(Float64(0.5 + Float64(0.5 * cos(Float64(v + v)))), Float64(e * Float64(-e)), 1.0))
end

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * N[(e * N[(1.0 - N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 + N[(0.5 * N[Cos[N[(v + v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(e * (-e)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 + e \cdot \cos v}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + e \cdot \cos v}} \]

   3. flip-+ N/A

      \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}{1 - e \cdot \cos v}}} \]

   4. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)} \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(e \cdot \sin v\right) \cdot \left(1 - e \cdot \cos v\right)}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(e \cdot \sin v\right) \cdot \left(1 - e \cdot \cos v\right)}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(e \cdot \sin v\right)} \cdot \left(1 - e \cdot \cos v\right)}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sin v \cdot e\right)} \cdot \left(1 - e \cdot \cos v\right)}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\sin v \cdot \color{blue}{\left(e \cdot \left(1 - e \cdot \cos v\right)\right)}}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\sin v \cdot \left(e \cdot \color{blue}{\left(1 - e \cdot \cos v\right)}\right)}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}{\color{blue}{1} - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \]

   14. sub-neg N/A

       \[\leadsto \frac{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}{\color{blue}{1 + \left(\mathsf{neg}\left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)\right)\right)}} \]

   15. +-commutative N/A

       \[\leadsto \frac{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)\right)\right) + 1}} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(v + v\right), -e \cdot e, 1\right)}} \]

5. Final simplification 99.9%

   \[\leadsto \frac{\sin v \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right)}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(v + v\right), e \cdot \left(-e\right), 1\right)} \]
6. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ e \cdot \left(\frac{\sin v}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(v + v\right), 0.5\right), e \cdot e, -1\right)} \cdot \mathsf{fma}\left(e, \cos v, -1\right)\right) \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (*
  e
  (*
   (/ (sin v) (fma (fma 0.5 (cos (+ v v)) 0.5) (* e e) -1.0))
   (fma e (cos v) -1.0))))

C

double code(double e, double v) {
	return e * ((sin(v) / fma(fma(0.5, cos((v + v)), 0.5), (e * e), -1.0)) * fma(e, cos(v), -1.0));
}

Julia

function code(e, v)
	return Float64(e * Float64(Float64(sin(v) / fma(fma(0.5, cos(Float64(v + v)), 0.5), Float64(e * e), -1.0)) * fma(e, cos(v), -1.0)))
end

Wolfram

code[e_, v_] := N[(e * N[(N[(N[Sin[v], $MachinePrecision] / N[(N[(0.5 * N[Cos[N[(v + v), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(e * e), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(e * N[Cos[v], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 + e \cdot \cos v}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e \cdot \sin v}}{1 + e \cdot \cos v} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]

   6. lower-/.f64 99.8

      \[\leadsto \color{blue}{\frac{\sin v}{1 + e \cdot \cos v}} \cdot e \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{1 + e \cdot \cos v}} \cdot e \]

   8. +-commutative N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v} + 1} \cdot e \]

   10. lower-fma.f64 99.8

       \[\leadsto \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\sin v}{\color{blue}{e \cdot \cos v} + 1} \cdot e \]

   4. flip-+ N/A

      \[\leadsto \frac{\sin v}{\color{blue}{\frac{\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot 1}{e \cdot \cos v - 1}}} \cdot e \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\left(\frac{\sin v}{\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot 1} \cdot \left(e \cdot \cos v - 1\right)\right)} \cdot e \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\sin v}{\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot 1} \cdot \left(e \cdot \cos v - 1\right)\right)} \cdot e \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\frac{\sin v}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(v + v\right), 0.5\right), e \cdot e, -1\right)} \cdot \mathsf{fma}\left(e, \cos v, -1\right)\right)} \cdot e \]

7. Final simplification 99.7%

   \[\leadsto e \cdot \left(\frac{\sin v}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(v + v\right), 0.5\right), e \cdot e, -1\right)} \cdot \mathsf{fma}\left(e, \cos v, -1\right)\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))