Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 13

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-/.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 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. *-lft-identity N/A

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

   5. associate-*r* N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 98.8% 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. 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. 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. unsub-neg N/A

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

   4. *-lft-identity N/A

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

   5. associate-*r* N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{e \cdot \left(\sin v \cdot \left(1 - e \cdot \cos v\right)\right)} \]
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 98.9

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

5. Applied rewrites 98.9%

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

6. Final simplification 98.9%

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

Alternative 5: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. *-lft-identity N/A

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

   5. associate-*r* N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in v around 0

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 6: 97.7% 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 98.6

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

5. Applied rewrites 98.6%

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

6. Final simplification 98.6%

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

Alternative 7: 51.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(e, -0.5, \mathsf{fma}\left(e, 0.16666666666666666, 0.16666666666666666\right)\right)\\ \frac{e}{\frac{\mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v \cdot v, e \cdot 0.041666666666666664 - \mathsf{fma}\left(-0.16666666666666666, t\_0, \mathsf{fma}\left(e, 0.008333333333333333, 0.008333333333333333\right)\right), t\_0\right), e + 1\right)}{v}} \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (fma e -0.5 (fma e 0.16666666666666666 0.16666666666666666))))
   (/
    e
    (/
     (fma
      (* v v)
      (fma
       (* v v)
       (-
        (* e 0.041666666666666664)
        (fma
         -0.16666666666666666
         t_0
         (fma e 0.008333333333333333 0.008333333333333333)))
       t_0)
      (+ e 1.0))
     v))))

C

double code(double e, double v) {
	double t_0 = fma(e, -0.5, fma(e, 0.16666666666666666, 0.16666666666666666));
	return e / (fma((v * v), fma((v * v), ((e * 0.041666666666666664) - fma(-0.16666666666666666, t_0, fma(e, 0.008333333333333333, 0.008333333333333333))), t_0), (e + 1.0)) / v);
}

Julia

function code(e, v)
	t_0 = fma(e, -0.5, fma(e, 0.16666666666666666, 0.16666666666666666))
	return Float64(e / Float64(fma(Float64(v * v), fma(Float64(v * v), Float64(Float64(e * 0.041666666666666664) - fma(-0.16666666666666666, t_0, fma(e, 0.008333333333333333, 0.008333333333333333))), t_0), Float64(e + 1.0)) / v))
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(e * -0.5 + N[(e * 0.16666666666666666 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(e / N[(N[(N[(v * v), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * N[(N[(e * 0.041666666666666664), $MachinePrecision] - N[(-0.16666666666666666 * t$95$0 + N[(e * 0.008333333333333333 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(e + 1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 55.4%

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

Alternative 8: 51.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, -0.5, \mathsf{fma}\left(e, 0.16666666666666666, 0.16666666666666666\right)\right), e + 1\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  e
  (/
   (fma
    v
    (* v (fma e -0.5 (fma e 0.16666666666666666 0.16666666666666666)))
    (+ e 1.0))
   v)))

C

double code(double e, double v) {
	return e / (fma(v, (v * fma(e, -0.5, fma(e, 0.16666666666666666, 0.16666666666666666))), (e + 1.0)) / v);
}

Julia

function code(e, v)
	return Float64(e / Float64(fma(v, Float64(v * fma(e, -0.5, fma(e, 0.16666666666666666, 0.16666666666666666))), Float64(e + 1.0)) / v))
end

Wolfram

code[e_, v_] := N[(e / N[(N[(v * N[(v * N[(e * -0.5 + N[(e * 0.16666666666666666 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(e + 1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 55.4%

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

Alternative 9: 50.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(v / N[(N[(v * N[(v * 0.16666666666666666), $MachinePrecision] + 1.0), $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

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 53.4%

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

10. Applied rewrites 53.3%

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

11. Taylor expanded in v around 0

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

13. Applied rewrites 55.0%

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

Alternative 10: 50.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(v * N[(e * N[(N[(e * e), $MachinePrecision] - 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 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 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 54.2%

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

Alternative 11: 50.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * N[(e - N[(e * 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 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 54.2%

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

Alternative 12: 50.0% 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 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. 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. unsub-neg N/A

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

   4. *-lft-identity N/A

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

   5. associate-*r* N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 54.2%

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

9. Final simplification 54.2%

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

Alternative 13: 49.4% 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 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.9%

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

9. Final simplification 53.9%

   \[\leadsto v \cdot e \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))