Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

Alternatives: 11

Speedup: N/A×

Specification

\[0 \leq e \land e \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cos-lowering-cos.f64 99.8

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

4. Applied egg-rr 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: 98.6% accurate, 1.9× speedup

Math

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

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

Derivation

1. Initial program 99.8%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sin-lowering-sin.f64 98.5

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

4. Applied egg-rr 98.5%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. cos-lowering-cos.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. +-lowering-+.f64 99.0

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

9. Simplified 99.0%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

    3. clear-num N/A

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

    4. *-lowering-*.f64 N/A

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

    5. /-lowering-/.f64 N/A

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

    6. +-lowering-+.f64 N/A

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

    7. sin-lowering-sin.f64 99.2

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

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

Alternative 3: 98.1% accurate, 2.0× speedup

Math

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

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(1.0 - e) * Float64(sin(v) * e))
end

MATLAB

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

Wolfram

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

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

   2. mul-1-neg N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

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

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

   9. distribute-rgt1-in N/A

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

   10. *-lowering-*.f64 N/A

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

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

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

   12. mul-1-neg N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(e, \color{blue}{0 - \cos v}, 1\right) \cdot \left(e \cdot \sin v\right) \]

   16. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(e, \color{blue}{0 - \cos v}, 1\right) \cdot \left(e \cdot \sin v\right) \]

   17. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(e, 0 - \color{blue}{\cos v}, 1\right) \cdot \left(e \cdot \sin v\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(e, 0 - \cos v, 1\right) \cdot \color{blue}{\left(e \cdot \sin v\right)} \]

5. Simplified 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e, 0 - \cos v, 1\right) \cdot \left(e \cdot \sin v\right)} \]

6. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 98.2

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 4: 74.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-27}:\\ \;\;\;\;e \cdot \frac{v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 2e-27) (* e (/ v (+ e 1.0))) (* (sin v) e)))

C

double code(double e, double v) {
	double tmp;
	if (v <= 2e-27) {
		tmp = e * (v / (e + 1.0));
	} else {
		tmp = sin(v) * e;
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 2d-27) then
        tmp = e * (v / (e + 1.0d0))
    else
        tmp = sin(v) * e
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 2e-27) {
		tmp = e * (v / (e + 1.0));
	} else {
		tmp = Math.sin(v) * e;
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 2e-27:
		tmp = e * (v / (e + 1.0))
	else:
		tmp = math.sin(v) * e
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 2e-27)
		tmp = Float64(e * Float64(v / Float64(e + 1.0)));
	else
		tmp = Float64(sin(v) * e);
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 2e-27)
		tmp = e * (v / (e + 1.0));
	else
		tmp = sin(v) * e;
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 2e-27], N[(e * N[(v / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2.0000000000000001e-27

   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 67.9

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

   5. Simplified 67.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 67.9

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

   7. Applied egg-rr 67.9%

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

   if 2.0000000000000001e-27 < v

   1. Initial program 99.7%

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

   3. Taylor expanded in e around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 98.2

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

   5. Simplified 98.2%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-27}:\\ \;\;\;\;e \cdot \frac{v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sin-lowering-sin.f64 98.5

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

7. Simplified 50.2%

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

   1. associate-/r/ N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. /-lowering-/.f64 51.4

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

9. Applied egg-rr 51.4%

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

10. Final simplification 51.4%

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

Alternative 6: 51.9% accurate, 4.6× speedup

Math

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * N[(e * -0.5 + N[(e * 0.16666666666666666 + 0.16666666666666666), $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. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. sin-lowering-sin.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. distribute-rgt-in N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 51.4

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

7. Simplified 51.4%

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

Alternative 7: 51.5% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(v * v), $MachinePrecision] * N[(e * -0.3333333333333333), $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. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. sin-lowering-sin.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. distribute-rgt-in N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 51.4

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

7. Simplified 51.4%

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

8. Taylor expanded in e around inf

   \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{\frac{-1}{3} \cdot e}, e + 1\right)}{v}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{e}{\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{e \cdot \frac{-1}{3}}, e + 1\right)}{v}} \]

   2. *-lowering-*.f64 51.2

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

10. Simplified 51.2%

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

Alternative 8: 50.9% accurate, 11.3× speedup

Math

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

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(e * Float64(v / Float64(e + 1.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 50.5

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

5. Simplified 50.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. +-lowering-+.f64 50.5

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

7. Applied egg-rr 50.5%

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

8. Final simplification 50.5%

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

Alternative 9: 50.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 50.5

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

5. Simplified 50.5%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 50.4

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

7. Applied egg-rr 50.4%

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

8. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f64 N/A

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

   5. *-lowering-*.f64 49.5

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

10. Simplified 49.5%

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

11. Final simplification 49.5%

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

Alternative 10: 49.8% 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. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 50.5

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

5. Simplified 50.5%

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

6. Taylor expanded in e around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 48.8

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

8. Simplified 48.8%

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

Alternative 11: 4.5% accurate, 225.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 v)

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return v

Julia

function code(e, v)
	return v
end

MATLAB

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

Wolfram

code[e_, v_] := v

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. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 50.5

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

5. Simplified 50.5%

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

6. Taylor expanded in e around inf

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

8. Simplified 4.8%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))