Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(e \cdot \sin v\_m\right)\right)}{1 + e \cdot \cos v\_m} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m)
 :precision binary64
 (* v_s (/ (log1p (expm1 (* e (sin v_m)))) (+ 1.0 (* e (cos v_m))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * (log1p(expm1((e * sin(v_m)))) / (1.0 + (e * cos(v_m))));
}

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * (Math.log1p(Math.expm1((e * Math.sin(v_m)))) / (1.0 + (e * Math.cos(v_m))));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * (math.log1p(math.expm1((e * math.sin(v_m)))) / (1.0 + (e * math.cos(v_m))))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(log1p(expm1(Float64(e * sin(v_m)))) / Float64(1.0 + Float64(e * cos(v_m)))))
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(N[Log[1 + N[(Exp[N[(e * N[Sin[v$95$m], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v$95$m], $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. log1p-expm1-u 99.8%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e \cdot \sin v\right)\right)}}{1 + e \cdot \cos v} \]

4. Applied egg-rr 99.8%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e \cdot \sin v\right)\right)}}{1 + e \cdot \cos v} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{e \cdot \sin v\_m}{1 + e \cdot \cos v\_m} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m)
 :precision binary64
 (* v_s (/ (* e (sin v_m)) (+ 1.0 (* e (cos v_m))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * ((e * sin(v_m)) / (1.0 + (e * cos(v_m))));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * ((e * sin(v_m)) / (1.0d0 + (e * cos(v_m))))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * ((e * Math.sin(v_m)) / (1.0 + (e * Math.cos(v_m))));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * ((e * math.sin(v_m)) / (1.0 + (e * math.cos(v_m))))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(Float64(e * sin(v_m)) / Float64(1.0 + Float64(e * cos(v_m)))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * ((e * sin(v_m)) / (1.0 + (e * cos(v_m))));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(N[(e * N[Sin[v$95$m], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v$95$m], $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. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{\sin v\_m}{\cos v\_m + \frac{1}{e}} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m)
 :precision binary64
 (* v_s (/ (sin v_m) (+ (cos v_m) (/ 1.0 e)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * (sin(v_m) / (cos(v_m) + (1.0 / e)));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * (sin(v_m) / (cos(v_m) + (1.0d0 / e)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * (Math.sin(v_m) / (Math.cos(v_m) + (1.0 / e)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * (math.sin(v_m) / (math.cos(v_m) + (1.0 / e)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(sin(v_m) / Float64(cos(v_m) + Float64(1.0 / e))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * (sin(v_m) / (cos(v_m) + (1.0 / e)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(N[Sin[v$95$m], $MachinePrecision] / N[(N[Cos[v$95$m], $MachinePrecision] + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in e around inf 99.7%

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

6. Taylor expanded in v around inf 99.6%

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

Alternative 4: 98.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{e \cdot \sin v\_m}{e + 1} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m) :precision binary64 (* v_s (/ (* e (sin v_m)) (+ e 1.0))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * ((e * sin(v_m)) / (e + 1.0));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * ((e * sin(v_m)) / (e + 1.0d0))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * ((e * Math.sin(v_m)) / (e + 1.0));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * ((e * math.sin(v_m)) / (e + 1.0))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(Float64(e * sin(v_m)) / Float64(e + 1.0)))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * ((e * sin(v_m)) / (e + 1.0));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(N[(e * N[Sin[v$95$m], $MachinePrecision]), $MachinePrecision] / 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 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(e \cdot \sin v\_m\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m) :precision binary64 (* v_s (* e (sin v_m))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * (e * sin(v_m));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * (e * sin(v_m))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * (e * Math.sin(v_m));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * (e * math.sin(v_m))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(e * sin(v_m)))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * (e * sin(v_m));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(e * N[Sin[v$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in e around 0 97.3%

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

Alternative 6: 51.8% accurate, 29.9× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(e \cdot \frac{v\_m}{e + 1}\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m) :precision binary64 (* v_s (* e (/ v_m (+ e 1.0)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * (e * (v_m / (e + 1.0)));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * (e * (v_m / (e + 1.0d0)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * (e * (v_m / (e + 1.0)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * (e * (v_m / (e + 1.0)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(e * Float64(v_m / Float64(e + 1.0))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * (e * (v_m / (e + 1.0)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(e * N[(v$95$m / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 56.9%

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

   1. associate-/l* 56.9%

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

   2. +-commutative 56.9%

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

7. Simplified 56.9%

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

Alternative 7: 51.4% accurate, 29.9× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(e \cdot \left(v\_m \cdot \left(1 - e\right)\right)\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m) :precision binary64 (* v_s (* e (* v_m (- 1.0 e)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * (e * (v_m * (1.0 - e)));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * (e * (v_m * (1.0d0 - e)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * (e * (v_m * (1.0 - e)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * (e * (v_m * (1.0 - e)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(e * Float64(v_m * Float64(1.0 - e))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * (e * (v_m * (1.0 - e)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(e * N[(v$95$m * N[(1.0 - e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 56.9%

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

   1. associate-/l* 56.9%

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

   2. +-commutative 56.9%

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

7. Simplified 56.9%

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

8. Taylor expanded in e around 0 56.5%

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

   1. neg-mul-1 56.5%

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

   2. *-lft-identity 56.5%

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

   3. distribute-lft-neg-in 56.5%

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

   4. distribute-rgt-in 56.5%

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

   5. sub-neg 56.5%

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

10. Simplified 56.5%

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

Alternative 8: 50.8% accurate, 69.7× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(e \cdot v\_m\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m) :precision binary64 (* v_s (* e v_m)))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * (e * v_m);
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * (e * v_m)
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * (e * v_m);
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * (e * v_m)

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * Float64(e * v_m))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * (e * v_m);
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * N[(e * v$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 56.9%

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

   1. associate-/l* 56.9%

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

   2. +-commutative 56.9%

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

7. Simplified 56.9%

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

8. Taylor expanded in e around 0 56.5%

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

   1. neg-mul-1 56.5%

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

   2. distribute-rgt-neg-in 56.5%

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

10. Simplified 56.5%

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

11. Taylor expanded in e around 0 55.9%

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

Alternative 9: 4.6% accurate, 209.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot v\_m \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s e v_m) :precision binary64 (* v_s v_m))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double e, double v_m) {
	return v_s * v_m;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, e, v_m)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: e
    real(8), intent (in) :: v_m
    code = v_s * v_m
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double e, double v_m) {
	return v_s * v_m;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, e, v_m):
	return v_s * v_m

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, e, v_m)
	return Float64(v_s * v_m)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, e, v_m)
	tmp = v_s * v_m;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, e_, v$95$m_] := N[(v$95$s * v$95$m), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. cos-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. sin-neg 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. sin-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. cos-neg 99.8%

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

   11. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 56.9%

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

   1. associate-/l* 56.9%

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

   2. +-commutative 56.9%

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

7. Simplified 56.9%

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

8. Taylor expanded in e around inf 4.8%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))