Trigonometry A

Percentage Accurate: 99.8% → 99.7%

Time: 8.2s

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 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $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. Final simplification 99.8%

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

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $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 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(e + 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. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[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 e around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Final simplification 98.1%

   \[\leadsto \sin v \cdot e \]
7. Add Preprocessing

Alternative 7: 51.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{v \cdot e}{1 + \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, -0.5, \left(e \cdot \left(v \cdot v\right)\right) \cdot \mathsf{fma}\left(v \cdot v, -0.001388888888888889, 0.041666666666666664\right)\right), e\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  (* v e)
  (+
   1.0
   (fma
    (* v v)
    (fma
     e
     -0.5
     (*
      (* e (* v v))
      (fma (* v v) -0.001388888888888889 0.041666666666666664)))
    e))))

C

double code(double e, double v) {
	return (v * e) / (1.0 + fma((v * v), fma(e, -0.5, ((e * (v * v)) * fma((v * v), -0.001388888888888889, 0.041666666666666664))), e));
}

Julia

function code(e, v)
	return Float64(Float64(v * e) / Float64(1.0 + fma(Float64(v * v), fma(e, -0.5, Float64(Float64(e * Float64(v * v)) * fma(Float64(v * v), -0.001388888888888889, 0.041666666666666664))), e)))
end

Wolfram

code[e_, v_] := N[(N[(v * e), $MachinePrecision] / N[(1.0 + N[(N[(v * v), $MachinePrecision] * N[(e * -0.5 + N[(N[(e * N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + e), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{-1}{720} \cdot \left(e \cdot {v}^{2}\right) + \frac{1}{24} \cdot e\right)\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\left({v}^{2} \cdot \left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{-1}{720} \cdot \left(e \cdot {v}^{2}\right) + \frac{1}{24} \cdot e\right)\right) + e\right)}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\mathsf{fma}\left({v}^{2}, \frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{-1}{720} \cdot \left(e \cdot {v}^{2}\right) + \frac{1}{24} \cdot e\right), e\right)}} \]

5. Applied rewrites 59.7%

   \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, -0.5, \left(e \cdot \left(v \cdot v\right)\right) \cdot \mathsf{fma}\left(v \cdot v, -0.001388888888888889, 0.041666666666666664\right)\right), e\right)}} \]

6. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, \frac{-1}{2}, \left(e \cdot \left(v \cdot v\right)\right) \cdot \mathsf{fma}\left(v \cdot v, \frac{-1}{720}, \frac{1}{24}\right)\right), e\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 55.4

      \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, -0.5, \left(e \cdot \left(v \cdot v\right)\right) \cdot \mathsf{fma}\left(v \cdot v, -0.001388888888888889, 0.041666666666666664\right)\right), e\right)} \]

8. Applied rewrites 55.4%

   \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, -0.5, \left(e \cdot \left(v \cdot v\right)\right) \cdot \mathsf{fma}\left(v \cdot v, -0.001388888888888889, 0.041666666666666664\right)\right), e\right)} \]

9. Final simplification 55.4%

   \[\leadsto \frac{v \cdot e}{1 + \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, -0.5, \left(e \cdot \left(v \cdot v\right)\right) \cdot \mathsf{fma}\left(v \cdot v, -0.001388888888888889, 0.041666666666666664\right)\right), e\right)} \]
10. Add Preprocessing

Alternative 8: 50.9% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 54.8

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

5. Applied rewrites 54.8%

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

7. Applied rewrites 54.8%

   \[\leadsto \frac{v \cdot e}{1 - e \cdot e} \cdot \color{blue}{\left(1 - e\right)} \]
8. Add Preprocessing

Alternative 9: 50.9% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 54.8

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

5. Applied rewrites 54.8%

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

7. Applied rewrites 54.8%

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

Alternative 10: 50.9% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * N[(e / N[(e + 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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 54.8

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

5. Applied rewrites 54.8%

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

6. Final simplification 54.8%

   \[\leadsto v \cdot \frac{e}{e + 1} \]
7. Add Preprocessing

Alternative 11: 50.6% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 54.8

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

5. Applied rewrites 54.8%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 54.5%

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

9. Final simplification 54.5%

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

Alternative 12: 50.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 54.8

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

5. Applied rewrites 54.8%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.4%

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

9. Taylor expanded in e around 0

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

11. Applied rewrites 54.2%

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

12. Final simplification 54.2%

    \[\leadsto e \cdot \left(v - v \cdot e\right) \]
13. Add Preprocessing

Alternative 13: 49.9% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * 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. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 54.8

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

5. Applied rewrites 54.8%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.4%

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

9. Final simplification 53.4%

   \[\leadsto v \cdot e \]
10. 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)))))