Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 98.3%

Time: 19.9s

Alternatives: 8

Speedup: 3.8×

• 34.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

Specification

Math

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

FPCore

(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))))

C

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));
}

Fortran

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

Java

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));
}

Python

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))

Julia

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

MATLAB

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

Wolfram

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]

Initial Program: 35.2% accurate, 1.0× speedup

Math

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

FPCore

(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))))

C

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));
}

Fortran

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

Java

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));
}

Python

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))

Julia

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

MATLAB

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

Wolfram

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]

Alternative 1: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\tan k \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot \sin k}{\frac{\ell}{k}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (tan k) (/ (* (* (/ k l) t) (sin k)) (/ l k)))))

C

double code(double t, double l, double k) {
	return 2.0 / (tan(k) * ((((k / l) * t) * sin(k)) / (l / k)));
}

Fortran

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 / (tan(k) * ((((k / l) * t) * sin(k)) / (l / k)))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / (math.tan(k) * ((((k / l) * t) * math.sin(k)) / (l / k)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(tan(k) * Float64(Float64(Float64(Float64(k / l) * t) * sin(k)) / Float64(l / k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (tan(k) * ((((k / l) * t) * sin(k)) / (l / k)));
end

Wolfram

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

Derivation

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Alternative 2: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{\frac{{k}^{3}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t}}{{k}^{2}}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.2e-122)
   (/ 2.0 (* (tan k) (/ (/ (pow k 3.0) (/ l t)) l)))
   (* l (* l (/ (/ (/ 2.0 t) (pow k 2.0)) (pow k 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.2e-122) {
		tmp = 2.0 / (tan(k) * ((pow(k, 3.0) / (l / t)) / l));
	} else {
		tmp = l * (l * (((2.0 / t) / pow(k, 2.0)) / pow(k, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.2d-122) then
        tmp = 2.0d0 / (tan(k) * (((k ** 3.0d0) / (l / t)) / l))
    else
        tmp = l * (l * (((2.0d0 / t) / (k ** 2.0d0)) / (k ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.2e-122) {
		tmp = 2.0 / (Math.tan(k) * ((Math.pow(k, 3.0) / (l / t)) / l));
	} else {
		tmp = l * (l * (((2.0 / t) / Math.pow(k, 2.0)) / Math.pow(k, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.2e-122:
		tmp = 2.0 / (math.tan(k) * ((math.pow(k, 3.0) / (l / t)) / l))
	else:
		tmp = l * (l * (((2.0 / t) / math.pow(k, 2.0)) / math.pow(k, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.2e-122)
		tmp = Float64(2.0 / Float64(tan(k) * Float64(Float64((k ^ 3.0) / Float64(l / t)) / l)));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(Float64(2.0 / t) / (k ^ 2.0)) / (k ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.2e-122)
		tmp = 2.0 / (tan(k) * (((k ^ 3.0) / (l / t)) / l));
	else
		tmp = l * (l * (((2.0 / t) / (k ^ 2.0)) / (k ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.2e-122], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(N[(2.0 / t), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 3: 92.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\tan k \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (tan k) (* (* t (sin k)) (* (/ k l) (/ k l))))))

C

double code(double t, double l, double k) {
	return 2.0 / (tan(k) * ((t * sin(k)) * ((k / l) * (k / l))));
}

Fortran

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 / (tan(k) * ((t * sin(k)) * ((k / l) * (k / l))))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / (math.tan(k) * ((t * math.sin(k)) * ((k / l) * (k / l))))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(tan(k) * Float64(Float64(t * sin(k)) * Float64(Float64(k / l) * Float64(k / l)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (tan(k) * ((t * sin(k)) * ((k / l) * (k / l))));
end

Wolfram

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

Derivation

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Alternative 4: 94.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\tan k \cdot \frac{k \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \sin k\right)\right)}{\ell}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (tan k) (/ (* k (* (/ k l) (* t (sin k)))) l))))

C

double code(double t, double l, double k) {
	return 2.0 / (tan(k) * ((k * ((k / l) * (t * sin(k)))) / l));
}

Fortran

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 / (tan(k) * ((k * ((k / l) * (t * sin(k)))) / l))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / (math.tan(k) * ((k * ((k / l) * (t * math.sin(k)))) / l))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(tan(k) * Float64(Float64(k * Float64(Float64(k / l) * Float64(t * sin(k)))) / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (tan(k) * ((k * ((k / l) * (t * sin(k)))) / l));
end

Wolfram

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

Derivation

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Alternative 5: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{\frac{{k}^{3}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{t}}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.8e+173)
   (/ 2.0 (* (tan k) (/ (/ (pow k 3.0) (/ l t)) l)))
   (* l (* l (/ (/ 2.0 t) (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.8e+173) {
		tmp = 2.0 / (tan(k) * ((pow(k, 3.0) / (l / t)) / l));
	} else {
		tmp = l * (l * ((2.0 / t) / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 5.8d+173) then
        tmp = 2.0d0 / (tan(k) * (((k ** 3.0d0) / (l / t)) / l))
    else
        tmp = l * (l * ((2.0d0 / t) / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.8e+173) {
		tmp = 2.0 / (Math.tan(k) * ((Math.pow(k, 3.0) / (l / t)) / l));
	} else {
		tmp = l * (l * ((2.0 / t) / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.8e+173:
		tmp = 2.0 / (math.tan(k) * ((math.pow(k, 3.0) / (l / t)) / l))
	else:
		tmp = l * (l * ((2.0 / t) / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.8e+173)
		tmp = Float64(2.0 / Float64(tan(k) * Float64(Float64((k ^ 3.0) / Float64(l / t)) / l)));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / t) / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.8e+173)
		tmp = 2.0 / (tan(k) * (((k ^ 3.0) / (l / t)) / l));
	else
		tmp = l * (l * ((2.0 / t) / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.8e+173], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(2.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 6: 70.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\tan k \cdot \frac{\frac{t \cdot {k}^{2}}{\ell}}{\frac{\ell}{k}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (tan k) (/ (/ (* t (pow k 2.0)) l) (/ l k)))))

C

double code(double t, double l, double k) {
	return 2.0 / (tan(k) * (((t * pow(k, 2.0)) / l) / (l / k)));
}

Fortran

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 / (tan(k) * (((t * (k ** 2.0d0)) / l) / (l / k)))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / (math.tan(k) * (((t * math.pow(k, 2.0)) / l) / (l / k)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(tan(k) * Float64(Float64(Float64(t * (k ^ 2.0)) / l) / Float64(l / k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (tan(k) * (((t * (k ^ 2.0)) / l) / (l / k)));
end

Wolfram

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

Derivation

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Alternative 7: 66.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(\ell \cdot \frac{\frac{2}{t}}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* l (/ (/ 2.0 t) (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return l * (l * ((2.0 / t) / pow(k, 4.0)));
}

Fortran

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

Java

public static double code(double t, double l, double k) {
	return l * (l * ((2.0 / t) / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return l * (l * ((2.0 / t) / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(l * Float64(l * Float64(Float64(2.0 / t) / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l * ((2.0 / t) / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l * N[(N[(2.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 8: 66.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (/ (/ (* t (pow k 4.0)) l) l)))

C

double code(double t, double l, double k) {
	return 2.0 / (((t * pow(k, 4.0)) / l) / l);
}

Fortran

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 * (k ** 4.0d0)) / l) / l)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((t * Math.pow(k, 4.0)) / l) / l);
}

Python

def code(t, l, k):
	return 2.0 / (((t * math.pow(k, 4.0)) / l) / l)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(t * (k ^ 4.0)) / l) / l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((t * (k ^ 4.0)) / l) / l);
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

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Reproduce

herbie shell --seed 2024010 
(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))))