Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.9s

Alternatives: 11

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

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

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.7

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

   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 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. lower-+.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 4: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

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

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

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{e}^{2}}\right)\right) \cdot \left(\cos v \cdot \sin v\right) + e \cdot \sin v \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left({e}^{2}\right)\right) \cdot \cos v\right) \cdot \sin v} + e \cdot \sin v \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right)} \cdot \sin v + e \cdot \sin v \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   12. lower-sin.f64 N/A

       \[\leadsto \color{blue}{\sin v} \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right) \]

   13. *-commutative N/A

       \[\leadsto \sin v \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\cos v \cdot {e}^{2}}\right)\right) + e\right) \]

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

       \[\leadsto \sin v \cdot \left(\color{blue}{\cos v \cdot \left(\mathsf{neg}\left({e}^{2}\right)\right)} + e\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto \sin v \cdot \color{blue}{\mathsf{fma}\left(\cos v, \mathsf{neg}\left({e}^{2}\right), e\right)} \]

5. Applied rewrites 98.5%

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

6. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

      \[\leadsto \sin v \cdot \left(e + \color{blue}{\left(\mathsf{neg}\left({e}^{2}\right)\right)}\right) \]

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 97.9

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

8. Applied rewrites 97.9%

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

Alternative 5: 75.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(e, -0.3333333333333333, 0.16666666666666666\right), e\right)\\ \mathbf{if}\;v \leq 6 \cdot 10^{-30}:\\ \;\;\;\;\frac{e \cdot v}{1 - t\_0 \cdot t\_0} \cdot \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (fma (* v v) (fma e -0.3333333333333333 0.16666666666666666) e)))
   (if (<= v 6e-30)
     (* (/ (* e v) (- 1.0 (* t_0 t_0))) (- 1.0 t_0))
     (* e (sin v)))))

C

double code(double e, double v) {
	double t_0 = fma((v * v), fma(e, -0.3333333333333333, 0.16666666666666666), e);
	double tmp;
	if (v <= 6e-30) {
		tmp = ((e * v) / (1.0 - (t_0 * t_0))) * (1.0 - t_0);
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Julia

function code(e, v)
	t_0 = fma(Float64(v * v), fma(e, -0.3333333333333333, 0.16666666666666666), e)
	tmp = 0.0
	if (v <= 6e-30)
		tmp = Float64(Float64(Float64(e * v) / Float64(1.0 - Float64(t_0 * t_0))) * Float64(1.0 - t_0));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(N[(v * v), $MachinePrecision] * N[(e * -0.3333333333333333 + 0.16666666666666666), $MachinePrecision] + e), $MachinePrecision]}, If[LessEqual[v, 6e-30], N[(N[(N[(e * v), $MachinePrecision] / N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 5.9999999999999998e-30

   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      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. lower-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. lower-*.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. lower-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. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-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. lower-+.f64 65.7

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

   7. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 65.8

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. associate-+r+ N/A

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

   9. Applied rewrites 65.8%

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(e \cdot v\right)} \cdot \frac{1}{1 + \left(v \cdot \left(v \cdot \left(e \cdot \frac{-1}{3} + \frac{1}{6}\right)\right) + e\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(e \cdot v\right)} \cdot \frac{1}{1 + \left(v \cdot \left(v \cdot \left(e \cdot \frac{-1}{3} + \frac{1}{6}\right)\right) + e\right)} \]

       3. lift-fma.f64 N/A

          \[\leadsto \left(e \cdot v\right) \cdot \frac{1}{1 + \left(v \cdot \left(v \cdot \color{blue}{\mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right)}\right) + e\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \left(e \cdot v\right) \cdot \frac{1}{1 + \left(v \cdot \color{blue}{\left(v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right)\right)} + e\right)} \]

       5. lift-fma.f64 N/A

          \[\leadsto \left(e \cdot v\right) \cdot \frac{1}{1 + \color{blue}{\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}} \]

       6. lift-+.f64 N/A

          \[\leadsto \left(e \cdot v\right) \cdot \frac{1}{\color{blue}{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}} \]

       7. clear-num N/A

          \[\leadsto \left(e \cdot v\right) \cdot \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}{1}}} \]

       8. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{e \cdot v}{\frac{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}{1}}} \]

       9. /-rgt-identity N/A

          \[\leadsto \frac{e \cdot v}{\color{blue}{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}} \]

       10. lift-+.f64 N/A

           \[\leadsto \frac{e \cdot v}{\color{blue}{1 + \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}} \]

       11. flip-+ N/A

           \[\leadsto \frac{e \cdot v}{\color{blue}{\frac{1 \cdot 1 - \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right) \cdot \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}{1 - \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)}}} \]

       12. associate-/r/ N/A

           \[\leadsto \color{blue}{\frac{e \cdot v}{1 \cdot 1 - \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right) \cdot \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)} \cdot \left(1 - \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \color{blue}{\frac{e \cdot v}{1 \cdot 1 - \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right) \cdot \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)} \cdot \left(1 - \mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(e, \frac{-1}{3}, \frac{1}{6}\right), e\right)\right)} \]

   11. Applied rewrites 64.9%

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

   if 5.9999999999999998e-30 < v

   1. Initial program 99.5%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 6: 52.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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 99.7

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

   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 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   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. lower-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. lower-*.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. lower-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. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. distribute-lft1-in N/A

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

   15. lower-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. lower-+.f64 54.5

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

7. Applied rewrites 54.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

9. Applied rewrites 54.6%

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

10. Final simplification 54.6%

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

Alternative 7: 51.8% 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 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 53.4

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

5. Applied rewrites 53.4%

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

   1. *-commutative N/A

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

   2. lift-+.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 53.4

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 53.4

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

7. Applied rewrites 53.4%

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

8. Final simplification 53.4%

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

Alternative 8: 51.2% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 53.4

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

5. Applied rewrites 53.4%

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

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. lower-*.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. lower--.f64 N/A

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

   5. lower-*.f64 52.2

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

8. Applied rewrites 52.2%

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

   1. lift-*.f64 N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 52.2

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

10. Applied rewrites 52.2%

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

11. Final simplification 52.2%

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

Alternative 9: 51.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in e around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

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

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

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{e}^{2}}\right)\right) \cdot \left(\cos v \cdot \sin v\right) + e \cdot \sin v \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left({e}^{2}\right)\right) \cdot \cos v\right) \cdot \sin v} + e \cdot \sin v \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right)} \cdot \sin v + e \cdot \sin v \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   12. lower-sin.f64 N/A

       \[\leadsto \color{blue}{\sin v} \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right) \]

   13. *-commutative N/A

       \[\leadsto \sin v \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\cos v \cdot {e}^{2}}\right)\right) + e\right) \]

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

       \[\leadsto \sin v \cdot \left(\color{blue}{\cos v \cdot \left(\mathsf{neg}\left({e}^{2}\right)\right)} + e\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto \sin v \cdot \color{blue}{\mathsf{fma}\left(\cos v, \mathsf{neg}\left({e}^{2}\right), e\right)} \]

5. Applied rewrites 98.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-neg.f64 N/A

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

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

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

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

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

   7. distribute-lft1-in N/A

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

   8. +-commutative N/A

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

   9. *-rgt-identity N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. cancel-sign-sub-inv N/A

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

   15. *-rgt-identity N/A

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

   16. +-commutative N/A

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

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

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

   18. lift-neg.f64 N/A

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

   19. lower-fma.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. mul-1-neg N/A

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

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

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

   9. unpow2 N/A

      \[\leadsto v \cdot \left(e + \left(\mathsf{neg}\left(\color{blue}{{e}^{2}}\right)\right)\right) \]

   10. unsub-neg N/A

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

   11. lower--.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 52.2

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

10. Applied rewrites 52.2%

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

Alternative 10: 51.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 53.4

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

5. Applied rewrites 53.4%

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

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. lower-*.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. lower--.f64 N/A

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

   5. lower-*.f64 52.2

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

8. Applied rewrites 52.2%

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

Alternative 11: 50.7% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in v around 0

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

   1. lower-*.f64 51.5

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

8. Applied rewrites 51.5%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))