Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.5s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.8

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

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in e around 0

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

   1. *-lft-identity N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 98.9

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

7. Simplified 98.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 98.9

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 4: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around 0

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

9. Simplified 98.9%

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

Alternative 5: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in v around 0

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] * N[(1.0 * (-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 e around 0

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 98.9

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

5. Simplified 98.9%

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

6. Taylor expanded in v around 0

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 7: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.1

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 8: 53.2% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

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. 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. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 53.1

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

7. Simplified 53.1%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

9. Applied egg-rr 53.2%

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

10. Final simplification 53.2%

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

Alternative 9: 52.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

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. 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. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 53.1

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

7. Simplified 53.1%

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

8. Taylor expanded in e around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 52.4

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

10. Simplified 52.4%

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

Alternative 10: 52.1% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * 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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 51.6

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

5. Simplified 51.6%

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

Alternative 11: 51.6% 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. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 51.6

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

5. Simplified 51.6%

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

6. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lft-identity N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

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

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

   11. *-lft-identity N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 51.1

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

8. Simplified 51.1%

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

9. Final simplification 51.1%

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

Alternative 12: 51.1% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 51.6

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

5. Simplified 51.6%

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

6. Taylor expanded in e around 0

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

   1. lower-*.f64 50.8

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

8. Simplified 50.8%

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

9. Final simplification 50.8%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))