Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[v], $MachinePrecision] * 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. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] * N[(N[Cos[v], $MachinePrecision] * (-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-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.6

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.3

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 98.7% accurate, 1.9× speedup

Math

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

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(Float64(e * 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[(N[(e * N[Sin[v], $MachinePrecision]), $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 99.2

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] * N[(1.0 - 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 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. Simplified 99.3%

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

6. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.8

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf

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

    1. sub-neg N/A

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

    2. distribute-rgt-in N/A

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

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

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

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

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

    7. mul-1-neg N/A

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

    8. distribute-lft-in N/A

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

    9. lower-*.f64 N/A

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

    10. associate-*r* N/A

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

    11. distribute-rgt1-in N/A

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

    12. +-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. mul-1-neg N/A

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

    15. unsub-neg N/A

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

    16. lower--.f64 N/A

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

    17. lower-sin.f64 98.8

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 5: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * 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. 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. Simplified 98.1%

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

Alternative 6: 51.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 98.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 98.8

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

4. Applied egg-rr 98.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. associate-/l/ N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 98.5

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 55.4

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

9. Simplified 55.4%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-+.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. un-div-inv N/A

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

11. Applied egg-rr 56.3%

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

12. Final simplification 56.3%

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

Alternative 7: 50.8% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 98.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 98.8

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

4. Applied egg-rr 98.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. associate-/l/ N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 98.5

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 55.4

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

9. Simplified 55.4%

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

10. Taylor expanded in e around 0

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

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{e \cdot v}{1 + \frac{1}{6} \cdot {v}^{2}}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{e \cdot v}}{1 + \frac{1}{6} \cdot {v}^{2}} \]

    3. +-commutative N/A

       \[\leadsto \frac{e \cdot v}{\color{blue}{\frac{1}{6} \cdot {v}^{2} + 1}} \]

    4. *-commutative N/A

       \[\leadsto \frac{e \cdot v}{\color{blue}{{v}^{2} \cdot \frac{1}{6}} + 1} \]

    5. unpow2 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 55.2

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

12. Simplified 55.2%

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

Alternative 8: 50.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * v), $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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 55.1

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

5. Simplified 55.1%

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

6. Final simplification 55.1%

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

Alternative 9: 50.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(e * N[(e * N[(N[(e * v), $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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 55.1

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

5. Simplified 55.1%

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

6. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 54.8

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

8. Simplified 54.8%

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

Alternative 10: 50.1% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v - N[(e * v), $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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 55.1

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

5. Simplified 55.1%

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

6. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. unpow2 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 54.6

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

8. Simplified 54.6%

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

9. Taylor expanded in v around 0

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

    1. sub-neg N/A

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

    2. distribute-rgt-in N/A

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

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

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

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

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

    7. mul-1-neg N/A

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

    8. distribute-lft-in N/A

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

    9. lower-*.f64 N/A

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

    10. mul-1-neg N/A

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

    11. unsub-neg N/A

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

    12. lower--.f64 N/A

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

    13. lower-*.f64 54.6

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

11. Simplified 54.6%

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

Alternative 11: 49.7% 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 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. Simplified 98.1%

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

6. Taylor expanded in v around 0

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

   1. lower-*.f64 54.0

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

8. Simplified 54.0%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))