Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 11

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   11. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 3: 52.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 + e}{e}\right)}^{-1} \cdot v \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* (pow (/ (+ 1.0 e) e) -1.0) v))

C

double code(double e, double v) {
	return pow(((1.0 + e) / e), -1.0) * v;
}

Fortran

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

Java

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

Python

def code(e, v):
	return math.pow(((1.0 + e) / e), -1.0) * v

Julia

function code(e, v)
	return Float64((Float64(Float64(1.0 + e) / e) ^ -1.0) * v)
end

MATLAB

function tmp = code(e, v)
	tmp = (((1.0 + e) / e) ^ -1.0) * v;
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

   \[\leadsto \color{blue}{\frac{e}{1 + e} \cdot v} \]
6. Step-by-step derivation

7. Applied rewrites 54.4%

   \[\leadsto \frac{1}{\frac{1 + e}{e}} \cdot v \]

8. Final simplification 54.4%

   \[\leadsto {\left(\frac{1 + e}{e}\right)}^{-1} \cdot v \]
9. Add Preprocessing

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + e}} \]
4. Step-by-step derivation

   1. lower-+.f64 98.6

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

5. Applied rewrites 98.6%

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + e}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\sin v}{1 + e} \cdot e} \]
8. Add Preprocessing

Alternative 5: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.0132:\\ \;\;\;\;\frac{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(0.008333333333333333, v \cdot v, -0.16666666666666666\right), v \cdot v, e\right) \cdot v}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(0.041666666666666664, v \cdot v, -0.5\right), v \cdot v, 1 + e\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 0.0132)
   (/
    (*
     (fma
      (* e (fma 0.008333333333333333 (* v v) -0.16666666666666666))
      (* v v)
      e)
     v)
    (fma (* e (fma 0.041666666666666664 (* v v) -0.5)) (* v v) (+ 1.0 e)))
   (* (sin v) e)))

C

double code(double e, double v) {
	double tmp;
	if (v <= 0.0132) {
		tmp = (fma((e * fma(0.008333333333333333, (v * v), -0.16666666666666666)), (v * v), e) * v) / fma((e * fma(0.041666666666666664, (v * v), -0.5)), (v * v), (1.0 + e));
	} else {
		tmp = sin(v) * e;
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 0.0132)
		tmp = Float64(Float64(fma(Float64(e * fma(0.008333333333333333, Float64(v * v), -0.16666666666666666)), Float64(v * v), e) * v) / fma(Float64(e * fma(0.041666666666666664, Float64(v * v), -0.5)), Float64(v * v), Float64(1.0 + e)));
	else
		tmp = Float64(sin(v) * e);
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 0.0132], N[(N[(N[(N[(e * N[(0.008333333333333333 * N[(v * v), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + e), $MachinePrecision] * v), $MachinePrecision] / N[(N[(e * N[(0.041666666666666664 * N[(v * v), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[(1.0 + e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 0.0132

   1. Initial program 99.9%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\left(1 + e\right) + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{e \cdot \sin v}{\color{blue}{{v}^{2} \cdot \left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right) + \left(1 + e\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right)\right) \cdot {v}^{2}} + \left(1 + e\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot e + \frac{1}{24} \cdot \left(e \cdot {v}^{2}\right), {v}^{2}, 1 + e\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot \left(e \cdot {v}^{2}\right) + \frac{-1}{2} \cdot e}, {v}^{2}, 1 + e\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\color{blue}{\left(e \cdot {v}^{2}\right) \cdot \frac{1}{24}} + \frac{-1}{2} \cdot e, {v}^{2}, 1 + e\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\color{blue}{e \cdot \left({v}^{2} \cdot \frac{1}{24}\right)} + \frac{-1}{2} \cdot e, {v}^{2}, 1 + e\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \left({v}^{2} \cdot \frac{1}{24}\right) + \color{blue}{e \cdot \frac{-1}{2}}, {v}^{2}, 1 + e\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\color{blue}{e \cdot \left({v}^{2} \cdot \frac{1}{24} + \frac{-1}{2}\right)}, {v}^{2}, 1 + e\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\color{blue}{e \cdot \left({v}^{2} \cdot \frac{1}{24} + \frac{-1}{2}\right)}, {v}^{2}, 1 + e\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \left(\color{blue}{\frac{1}{24} \cdot {v}^{2}} + \frac{-1}{2}\right), {v}^{2}, 1 + e\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {v}^{2}, \frac{-1}{2}\right)}, {v}^{2}, 1 + e\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, \color{blue}{v \cdot v}, \frac{-1}{2}\right), {v}^{2}, 1 + e\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, \color{blue}{v \cdot v}, \frac{-1}{2}\right), {v}^{2}, 1 + e\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, v \cdot v, \frac{-1}{2}\right), \color{blue}{v \cdot v}, 1 + e\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, v \cdot v, \frac{-1}{2}\right), \color{blue}{v \cdot v}, 1 + e\right)} \]

      17. lower-+.f64 72.8

          \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(0.041666666666666664, v \cdot v, -0.5\right), v \cdot v, \color{blue}{1 + e}\right)} \]

   5. Applied rewrites 72.8%

      \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(0.041666666666666664, v \cdot v, -0.5\right), v \cdot v, 1 + e\right)}} \]

   6. Taylor expanded in v around 0

      \[\leadsto \frac{\color{blue}{v \cdot \left(e + {v}^{2} \cdot \left(\frac{-1}{6} \cdot e + \frac{1}{120} \cdot \left(e \cdot {v}^{2}\right)\right)\right)}}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, v \cdot v, \frac{-1}{2}\right), v \cdot v, 1 + e\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(e + {v}^{2} \cdot \left(\frac{-1}{6} \cdot e + \frac{1}{120} \cdot \left(e \cdot {v}^{2}\right)\right)\right) \cdot v}}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, v \cdot v, \frac{-1}{2}\right), v \cdot v, 1 + e\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(e + {v}^{2} \cdot \left(\frac{-1}{6} \cdot e + \frac{1}{120} \cdot \left(e \cdot {v}^{2}\right)\right)\right) \cdot v}}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(\frac{1}{24}, v \cdot v, \frac{-1}{2}\right), v \cdot v, 1 + e\right)} \]

   8. Applied rewrites 69.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(0.008333333333333333, v \cdot v, -0.16666666666666666\right), v \cdot v, e\right) \cdot v}}{\mathsf{fma}\left(e \cdot \mathsf{fma}\left(0.041666666666666664, v \cdot v, -0.5\right), v \cdot v, 1 + e\right)} \]

   if 0.0132 < v

   1. Initial program 99.7%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   2. Add Preprocessing

   3. Taylor expanded in e around 0

      \[\leadsto \color{blue}{e \cdot \sin v} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin v \cdot e} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin v \cdot e} \]

      3. lower-sin.f64 97.4

         \[\leadsto \color{blue}{\sin v} \cdot e \]

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\sin v \cdot e} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{\mathsf{fma}\left(0.16666666666666666, v \cdot v, 1 + e\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/ e (/ (fma 0.16666666666666666 (* v v) (+ 1.0 e)) v)))

C

double code(double e, double v) {
	return e / (fma(0.16666666666666666, (v * v), (1.0 + e)) / v);
}

Julia

function code(e, v)
	return Float64(e / Float64(fma(0.16666666666666666, Float64(v * v), Float64(1.0 + e)) / v))
end

Wolfram

code[e_, v_] := N[(e / N[(N[(0.16666666666666666 * N[(v * v), $MachinePrecision] + N[(1.0 + e), $MachinePrecision]), $MachinePrecision] / v), $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. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.6

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{e}{\color{blue}{\frac{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{e}{\color{blue}{\frac{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}}} \]

   2. associate-+r+ N/A

      \[\leadsto \frac{e}{\frac{\color{blue}{\left(1 + e\right) + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)}}{v}} \]

   3. +-commutative N/A

      \[\leadsto \frac{e}{\frac{\color{blue}{{v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \left(1 + e\right)}}{v}} \]

   4. *-commutative N/A

      \[\leadsto \frac{e}{\frac{\color{blue}{\left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) \cdot {v}^{2}} + \left(1 + e\right)}{v}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{e}{\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right), {v}^{2}, 1 + e\right)}}{v}} \]

   6. sub-neg N/A

      \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot e + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}, {v}^{2}, 1 + e\right)}{v}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, e, \mathsf{neg}\left(\frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}, {v}^{2}, 1 + e\right)}{v}} \]

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

      \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \left(1 + e\right)}\right), {v}^{2}, 1 + e\right)}{v}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \color{blue}{\frac{1}{6}} \cdot \left(1 + e\right)\right), {v}^{2}, 1 + e\right)}{v}} \]

   10. +-commutative N/A

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \frac{1}{6} \cdot \color{blue}{\left(e + 1\right)}\right), {v}^{2}, 1 + e\right)}{v}} \]

   11. distribute-lft-in N/A

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \color{blue}{\frac{1}{6} \cdot e + \frac{1}{6} \cdot 1}\right), {v}^{2}, 1 + e\right)}{v}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \frac{1}{6} \cdot e + \color{blue}{\frac{1}{6}}\right), {v}^{2}, 1 + e\right)}{v}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \color{blue}{\mathsf{fma}\left(\frac{1}{6}, e, \frac{1}{6}\right)}\right), {v}^{2}, 1 + e\right)}{v}} \]

   14. unpow2 N/A

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \mathsf{fma}\left(\frac{1}{6}, e, \frac{1}{6}\right)\right), \color{blue}{v \cdot v}, 1 + e\right)}{v}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, e, \mathsf{fma}\left(\frac{1}{6}, e, \frac{1}{6}\right)\right), \color{blue}{v \cdot v}, 1 + e\right)}{v}} \]

   16. lower-+.f64 55.4

       \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right), v \cdot v, \color{blue}{1 + e}\right)}{v}} \]

7. Applied rewrites 55.4%

   \[\leadsto \frac{e}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right), v \cdot v, 1 + e\right)}{v}}} \]

8. Taylor expanded in e around 0

   \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(\frac{1}{6}, v \cdot v, 1 + e\right)}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 55.4%

    \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(0.16666666666666666, v \cdot v, 1 + e\right)}{v}} \]
11. Add Preprocessing

Alternative 7: 52.1% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return (v / (1.0 + e)) * e

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v / N[(1.0 + e), $MachinePrecision]), $MachinePrecision] * e), $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. *-commutative N/A

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

   11. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{v}{1 + e}} \cdot e \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-+.f64 54.4

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

7. Applied rewrites 54.4%

   \[\leadsto \color{blue}{\frac{v}{1 + e}} \cdot e \]
8. Add Preprocessing

Alternative 8: 52.1% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return (e / (1.0 + e)) * v

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

   \[\leadsto \color{blue}{\frac{e}{1 + e} \cdot v} \]
6. Add Preprocessing

Alternative 9: 51.8% accurate, 11.3× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 (* (* (fma (- e 1.0) e 1.0) e) v))

C

double code(double e, double v) {
	return (fma((e - 1.0), e, 1.0) * e) * v;
}

Julia

function code(e, v)
	return Float64(Float64(fma(Float64(e - 1.0), e, 1.0) * e) * v)
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in e around 0

   \[\leadsto \left(e \cdot \left(1 + e \cdot \left(e - 1\right)\right)\right) \cdot v \]
7. Step-by-step derivation

8. Applied rewrites 53.8%

   \[\leadsto \left(\mathsf{fma}\left(e - 1, e, 1\right) \cdot e\right) \cdot v \]
9. Add Preprocessing

Alternative 10: 51.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-v, e, v\right) \cdot e \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* (fma (- v) e v) e))

C

double code(double e, double v) {
	return fma(-v, e, v) * e;
}

Julia

function code(e, v)
	return Float64(fma(Float64(-v), e, v) * e)
end

Wolfram

code[e_, v_] := N[(N[((-v) * e + v), $MachinePrecision] * e), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in e around 0

   \[\leadsto e \cdot \color{blue}{\left(v + -1 \cdot \left(e \cdot v\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 53.6%

   \[\leadsto \mathsf{fma}\left(-v, e, v\right) \cdot \color{blue}{e} \]
9. Add Preprocessing

Alternative 11: 51.1% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ e \cdot v \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e v))

C

double code(double e, double v) {
	return e * v;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * v
end function

Java

public static double code(double e, double v) {
	return e * v;
}

Python

def code(e, v):
	return e * v

Julia

function code(e, v)
	return Float64(e * v)
end

MATLAB

function tmp = code(e, v)
	tmp = e * v;
end

Wolfram

code[e_, v_] := N[(e * v), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in e around 0

   \[\leadsto e \cdot \color{blue}{v} \]
7. Step-by-step derivation

8. Applied rewrites 53.0%

   \[\leadsto e \cdot \color{blue}{v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))