Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 85.3%
Time: 15.1s
Alternatives: 4
Speedup: 17.3×

Specification

?
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t_1 \cdot t_1\\ \frac{\frac{t_0 \cdot \left(\beta \cdot \alpha + t_0\right)}{t_2}}{t_2 - 1} \end{array} \end{array} \]
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))
double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\
t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\
t_2 := t_1 \cdot t_1\\
\frac{\frac{t_0 \cdot \left(\beta \cdot \alpha + t_0\right)}{t_2}}{t_2 - 1}
\end{array}
\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 4 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: 16.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t_1 \cdot t_1\\ \frac{\frac{t_0 \cdot \left(\beta \cdot \alpha + t_0\right)}{t_2}}{t_2 - 1} \end{array} \end{array} \]
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))
double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\
t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\
t_2 := t_1 \cdot t_1\\
\frac{\frac{t_0 \cdot \left(\beta \cdot \alpha + t_0\right)}{t_2}}{t_2 - 1}
\end{array}
\end{array}

Alternative 1: 85.3% accurate, 4.8× speedup?

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+167}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]
NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.25e+167) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))
assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.25e+167) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.25d+167) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.25e+167) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.25e+167:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.25e+167)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.25e+167)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.25e+167], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
[alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\
\\
\begin{array}{l}
\mathbf{if}\;\beta \leq 2.25 \cdot 10^{+167}:\\
\;\;\;\;0.0625\\

\mathbf{else}:\\
\;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 75.3% accurate, 5.8× speedup?

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+226}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \alpha}{\beta}\\ \end{array} \end{array} \]
NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.2e+226) 0.0625 (/ (* (/ i beta) alpha) beta)))
assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+226) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * alpha) / beta;
	}
	return tmp;
}

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.2d+226) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * alpha) / beta
    end if
    code = tmp
end function

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+226) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * alpha) / beta;
	}
	return tmp;
}

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.2e+226:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * alpha) / beta
	return tmp

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.2e+226)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * alpha) / beta);
	end
	return tmp
end

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.2e+226)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * alpha) / beta;
	end
	tmp_2 = tmp;
end

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.2e+226], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * alpha), $MachinePrecision] / beta), $MachinePrecision]]

\begin{array}{l}
[alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\
\\
\begin{array}{l}
\mathbf{if}\;\beta \leq 2.2 \cdot 10^{+226}:\\
\;\;\;\;0.0625\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{i}{\beta} \cdot \alpha}{\beta}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 74.2% accurate, 17.3× speedup?

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+226}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]
NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 (if (<= beta 3.2e+226) 0.0625 0.0))
assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.2e+226) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.2d+226) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.2e+226) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.2e+226:
		tmp = 0.0625
	else:
		tmp = 0.0
	return tmp

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.2e+226)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	return tmp
end

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.2e+226)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.2e+226], 0.0625, 0.0]

\begin{array}{l}
[alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\
\\
\begin{array}{l}
\mathbf{if}\;\beta \leq 3.2 \cdot 10^{+226}:\\
\;\;\;\;0.0625\\

\mathbf{else}:\\
\;\;\;\;0\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 10.2% accurate, 53.0× speedup?

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0 \end{array} \]
NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0)
assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return 0.0;
}

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0d0
end function

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return 0.0;
}

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0
end

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0;
end

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0

\begin{array}{l}
[alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\
\\
0
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2024008 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))