Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   11. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. lower-*.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-lft-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-sin.f64 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around inf

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

   1. rgt-mult-inverse N/A

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

   2. distribute-lft-in N/A

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

   3. +-commutative N/A

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

   4. times-frac N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. rgt-mult-inverse N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-/.f64 99.7

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

5. Applied rewrites 99.7%

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in e around 0

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 4: 97.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. *-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 97.9

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 5: 52.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.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 99.6

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

   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. *-commutative N/A

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

   12. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

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

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

   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. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 55.6

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

7. Applied rewrites 55.6%

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

8. Final simplification 55.6%

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

Alternative 6: 51.3% 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. +-commutative N/A

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

   5. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Final simplification 54.4%

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

Alternative 7: 50.7% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v - N[(e * 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 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. +-commutative N/A

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

   5. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.7%

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

Alternative 8: 50.2% 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. +-commutative N/A

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

   5. lower-+.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.3%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))