Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 91.9%
Time: 17.9s
Alternatives: 9
Speedup: 2.0×

Specification

?
\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 9 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 34.4% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}
\end{array}

Alternative 1: 91.9% accurate, 0.8× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{{\left(\sin k \cdot \sqrt{t_m}\right)}^{2}}{\cos k}} \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (/
   2.0
   (*
    (pow (* k (/ 1.0 l)) 2.0)
    (/ (pow (* (sin k) (sqrt t_m)) 2.0) (cos k))))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow((k * (1.0 / l)), 2.0) * (pow((sin(k) * sqrt(t_m)), 2.0) / cos(k))));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((k * (1.0d0 / l)) ** 2.0d0) * (((sin(k) * sqrt(t_m)) ** 2.0d0) / cos(k))))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (Math.pow((k * (1.0 / l)), 2.0) * (Math.pow((Math.sin(k) * Math.sqrt(t_m)), 2.0) / Math.cos(k))));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow((k * (1.0 / l)), 2.0) * (math.pow((math.sin(k) * math.sqrt(t_m)), 2.0) / math.cos(k))))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((Float64(k * Float64(1.0 / l)) ^ 2.0) * Float64((Float64(sin(k) * sqrt(t_m)) ^ 2.0) / cos(k)))))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((k * (1.0 / l)) ^ 2.0) * (((sin(k) * sqrt(t_m)) ^ 2.0) / cos(k))));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[Power[N[(k * N[(1.0 / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{{\left(\sin k \cdot \sqrt{t_m}\right)}^{2}}{\cos k}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 93.1% accurate, 1.0× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\frac{\cos k}{t_m} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}\right) \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ (cos k) t_m) (pow (* (/ l k) (/ (sqrt 2.0) (sin k))) 2.0))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((cos(k) / t_m) * pow(((l / k) * (sqrt(2.0) / sin(k))), 2.0));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((cos(k) / t_m) * (((l / k) * (sqrt(2.0d0) / sin(k))) ** 2.0d0))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((Math.cos(k) / t_m) * Math.pow(((l / k) * (Math.sqrt(2.0) / Math.sin(k))), 2.0));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((math.cos(k) / t_m) * math.pow(((l / k) * (math.sqrt(2.0) / math.sin(k))), 2.0))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(cos(k) / t_m) * (Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) ^ 2.0)))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((cos(k) / t_m) * (((l / k) * (sqrt(2.0) / sin(k))) ^ 2.0));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \left(\frac{\cos k}{t_m} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 90.8% accurate, 1.3× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t_m \cdot {\sin k}^{2}}{\cos k}} \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (/ k l) (/ k l)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k / l) * (k / l)) * ((t_m * pow(sin(k), 2.0)) / cos(k))));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((k / l) * (k / l)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k))))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k / l) * (k / l)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((k / l) * (k / l)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k))))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k)))))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((k / l) * (k / l)) * ((t_m * (sin(k) ^ 2.0)) / cos(k))));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t_m \cdot {\sin k}^{2}}{\cos k}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 70.7% accurate, 1.3× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t_m \cdot {k}^{2}}{\cos k}} \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (pow (* k (/ 1.0 l)) 2.0) (/ (* t_m (pow k 2.0)) (cos k))))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow((k * (1.0 / l)), 2.0) * ((t_m * pow(k, 2.0)) / cos(k))));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((k * (1.0d0 / l)) ** 2.0d0) * ((t_m * (k ** 2.0d0)) / cos(k))))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (Math.pow((k * (1.0 / l)), 2.0) * ((t_m * Math.pow(k, 2.0)) / Math.cos(k))));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow((k * (1.0 / l)), 2.0) * ((t_m * math.pow(k, 2.0)) / math.cos(k))))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((Float64(k * Float64(1.0 / l)) ^ 2.0) * Float64(Float64(t_m * (k ^ 2.0)) / cos(k)))))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((k * (1.0 / l)) ^ 2.0) * ((t_m * (k ^ 2.0)) / cos(k))));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[Power[N[(k * N[(1.0 / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t_m \cdot {k}^{2}}{\cos k}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 71.1% accurate, 1.4× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot {\left(\frac{\ell}{\sqrt{t_m}} \cdot {k}^{-2}\right)}^{2}\right) \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (pow (* (/ l (sqrt t_m)) (pow k -2.0)) 2.0))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * pow(((l / sqrt(t_m)) * pow(k, -2.0)), 2.0));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l / sqrt(t_m)) * (k ** (-2.0d0))) ** 2.0d0))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * Math.pow(((l / Math.sqrt(t_m)) * Math.pow(k, -2.0)), 2.0));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * math.pow(((l / math.sqrt(t_m)) * math.pow(k, -2.0)), 2.0))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * (Float64(Float64(l / sqrt(t_m)) * (k ^ -2.0)) ^ 2.0)))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l / sqrt(t_m)) * (k ^ -2.0)) ^ 2.0));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[Power[N[(N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \left(2 \cdot {\left(\frac{\ell}{\sqrt{t_m}} \cdot {k}^{-2}\right)}^{2}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 59.0% accurate, 2.0× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \left({k}^{-4} \cdot \frac{{\ell}^{2}}{t_m}\right)\right) \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (* (pow k -4.0) (/ (pow l 2.0) t_m)))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (pow(k, -4.0) * (pow(l, 2.0) / t_m)));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((k ** (-4.0d0)) * ((l ** 2.0d0) / t_m)))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (Math.pow(k, -4.0) * (Math.pow(l, 2.0) / t_m)));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (math.pow(k, -4.0) * (math.pow(l, 2.0) / t_m)))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((k ^ -4.0) * Float64((l ^ 2.0) / t_m))))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((k ^ -4.0) * ((l ^ 2.0) / t_m)));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[Power[k, -4.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \left(2 \cdot \left({k}^{-4} \cdot \frac{{\ell}^{2}}{t_m}\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 60.3% accurate, 2.0× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t_m}\right) \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t_m))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / t_m));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t_m))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) * Math.pow(k, -4.0)) / t_m));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) * math.pow(k, -4.0)) / t_m))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) * (k ^ -4.0)) / t_m)))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) * (k ^ -4.0)) / t_m));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \left(2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t_m}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 60.3% accurate, 2.0× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t_m \cdot {k}^{4}}\right) \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0))));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0))))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0))));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0))))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0)))))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0))));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t_m \cdot {k}^{4}}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 60.3% accurate, 2.0× speedup?

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\frac{2}{t_m} \cdot \frac{{\ell}^{2}}{{k}^{4}}\right) \end{array} \]
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k 4.0)))))
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / t_m) * (pow(l, 2.0) / pow(k, 4.0)));
}

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 / t_m) * ((l ** 2.0d0) / (k ** 4.0d0)))
end function

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k, 4.0)));
}

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k, 4.0)))

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k ^ 4.0))))
end

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 / t_m) * ((l ^ 2.0) / (k ^ 4.0)));
end

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
t_m = \left|t\right|
\\
t_s = \mathsf{copysign}\left(1, t\right)

\\
t_s \cdot \left(\frac{2}{t_m} \cdot \frac{{\ell}^{2}}{{k}^{4}}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2024003 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))