Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 10

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos v \cdot e\\ \frac{\sin v \cdot e}{{t\_0}^{2} - 1} \cdot \left(t\_0 - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (* (cos v) e)))
   (* (/ (* (sin v) e) (- (pow t_0 2.0) 1.0)) (- t_0 1.0))))

C

double code(double e, double v) {
	double t_0 = cos(v) * e;
	return ((sin(v) * e) / (pow(t_0, 2.0) - 1.0)) * (t_0 - 1.0);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: t_0
    t_0 = cos(v) * e
    code = ((sin(v) * e) / ((t_0 ** 2.0d0) - 1.0d0)) * (t_0 - 1.0d0)
end function

Java

public static double code(double e, double v) {
	double t_0 = Math.cos(v) * e;
	return ((Math.sin(v) * e) / (Math.pow(t_0, 2.0) - 1.0)) * (t_0 - 1.0);
}

Python

def code(e, v):
	t_0 = math.cos(v) * e
	return ((math.sin(v) * e) / (math.pow(t_0, 2.0) - 1.0)) * (t_0 - 1.0)

Julia

function code(e, v)
	t_0 = Float64(cos(v) * e)
	return Float64(Float64(Float64(sin(v) * e) / Float64((t_0 ^ 2.0) - 1.0)) * Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(e, v)
	t_0 = cos(v) * e;
	tmp = ((sin(v) * e) / ((t_0 ^ 2.0) - 1.0)) * (t_0 - 1.0);
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision]}, N[(N[(N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\sin v \cdot e}{{\left(\cos v \cdot e\right)}^{2} - 1} \cdot \left(\cos v \cdot e - 1\right)} \]
5. Add Preprocessing

Alternative 2: 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]

Derivation

1. Initial program 99.8%

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

Alternative 3: 98.8% accurate, 1.9× speedup

Math

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

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(Float64(e * 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[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + e), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. lower-+.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 4: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 98.6%

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

Alternative 5: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 98.2

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

5. Applied rewrites 98.2%

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

Alternative 6: 51.7% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 47.8

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

5. Applied rewrites 47.8%

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

7. Applied rewrites 47.8%

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

Alternative 7: 51.7% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 47.8

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

5. Applied rewrites 47.8%

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

7. Applied rewrites 47.8%

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

Alternative 8: 51.7% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 47.8

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

5. Applied rewrites 47.8%

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

Alternative 9: 51.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 46.7%

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

Alternative 10: 50.7% 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 v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 47.8

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

5. Applied rewrites 47.8%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 46.4%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))