Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 8

Speedup: 1.0×

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

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

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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. 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 2: 98.9% accurate, 1.0× speedup

Math

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

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(e * Float64(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[(e * N[(N[Sin[v], $MachinePrecision] * N[(1.0 - N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. 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 e around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

   \[\leadsto e \cdot \left(\sin v \cdot \left(1 - e \cdot \cos v\right)\right) \]
9. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(N[Cos[v], $MachinePrecision] * N[(e * (-e)), $MachinePrecision] + 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 e around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{e}^{2}}\right)\right) \cdot \left(\cos v \cdot \sin v\right) + e \cdot \sin v \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left({e}^{2}\right)\right) \cdot \cos v\right) \cdot \sin v} + e \cdot \sin v \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right)} \cdot \sin v + e \cdot \sin v \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   12. lower-sin.f64 N/A

       \[\leadsto \color{blue}{\sin v} \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right) \]

   13. *-commutative N/A

       \[\leadsto \sin v \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\cos v \cdot {e}^{2}}\right)\right) + e\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \sin v \cdot \left(\color{blue}{\cos v \cdot \left(\mathsf{neg}\left({e}^{2}\right)\right)} + e\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto \sin v \cdot \color{blue}{\mathsf{fma}\left(\cos v, \mathsf{neg}\left({e}^{2}\right), e\right)} \]

5. Applied rewrites 99.3%

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

Alternative 4: 98.9% 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 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. Final simplification 98.6%

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

Alternative 5: 97.9% 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 97.9

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 6: 52.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{1}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, -0.3333333333333333, 0.16666666666666666\right), e\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (*
  e
  (/
   1.0
   (/
    (+ 1.0 (fma v (* v (fma e -0.3333333333333333 0.16666666666666666)) e))
    v))))

C

double code(double e, double v) {
	return e * (1.0 / ((1.0 + fma(v, (v * fma(e, -0.3333333333333333, 0.16666666666666666)), e)) / v));
}

Julia

function code(e, v)
	return Float64(e * Float64(1.0 / Float64(Float64(1.0 + fma(v, Float64(v * fma(e, -0.3333333333333333, 0.16666666666666666)), e)) / v)))
end

Wolfram

code[e_, v_] := N[(e * N[(1.0 / N[(N[(1.0 + N[(v * N[(v * N[(e * -0.3333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + e), $MachinePrecision]), $MachinePrecision] / v), $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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. 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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-approx N/A

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

   9. lift-*.f64 99.3

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

8. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, -0.3333333333333333, 0.16666666666666666\right), e\right)}{v}} \cdot e} \]

9. Final simplification 50.8%

   \[\leadsto e \cdot \frac{1}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, -0.3333333333333333, 0.16666666666666666\right), e\right)}{v}} \]
10. Add Preprocessing

Alternative 7: 52.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, -0.3333333333333333, 0.16666666666666666\right), e\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  e
  (/
   (+ 1.0 (fma v (* v (fma e -0.3333333333333333 0.16666666666666666)) e))
   v)))

C

double code(double e, double v) {
	return e / ((1.0 + fma(v, (v * fma(e, -0.3333333333333333, 0.16666666666666666)), e)) / v);
}

Julia

function code(e, v)
	return Float64(e / Float64(Float64(1.0 + fma(v, Float64(v * fma(e, -0.3333333333333333, 0.16666666666666666)), e)) / v))
end

Wolfram

code[e_, v_] := N[(e / N[(N[(1.0 + N[(v * N[(v * N[(e * -0.3333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. 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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-*l/ N/A

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

   6. associate-/r/ N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-approx 50.8

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

8. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, -0.3333333333333333, 0.16666666666666666\right), e\right)}{v}}} \]
9. Add Preprocessing

Alternative 8: 51.5% 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. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
6. 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. +-commutative N/A

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

   6. lower-+.f64 49.7

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

7. Applied rewrites 49.7%

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

Reproduce

herbie shell --seed 2024212 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))