Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.1% → 55.2%
Time: 28.3s
Alternatives: 5
Speedup: 151.0×

Specification

?
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \end{array} \end{array} \]
(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\
\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)
\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 5 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: 24.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \end{array} \end{array} \]
(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\
\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)
\end{array}
\end{array}

Alternative 1: 55.2% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{\frac{D}{d} \cdot \left(\left(w \cdot h\right) \cdot \frac{D}{d}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]
(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* 2.0 (/ c0 (* (/ D d) (* (* w h) (/ D d))))))
     0.0)))
double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (c0 / ((D / d) * ((w * h) * (D / d)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * (c0 / ((D / d) * ((w * h) * (D / d)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (2.0 * (c0 / ((D / d) * ((w * h) * (D / d)))))
	else:
		tmp = 0.0
	return tmp

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(Float64(D / d) * Float64(Float64(w * h) * Float64(D / d))))));
	else
		tmp = 0.0;
	end
	return tmp
end

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (2.0 * (c0 / ((D / d) * ((w * h) * (D / d)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(c0 / N[(N[(D / d), $MachinePrecision] * N[(N[(w * h), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{c0}{2 \cdot w}\\
t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\
\mathbf{if}\;t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq \infty:\\
\;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{\frac{D}{d} \cdot \left(\left(w \cdot h\right) \cdot \frac{D}{d}\right)}\right)\\

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


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

Alternative 2: 39.1% accurate, 4.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.05 \cdot 10^{-153} \lor \neg \left(M \leq 1.85 \cdot 10^{-59} \lor \neg \left(M \leq 4.8 \cdot 10^{-38}\right) \land M \leq 1.55 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]
(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 4.1e-171)
   0.0
   (if (or (<= M 2.05e-153)
           (not
            (or (<= M 1.85e-59) (and (not (<= M 4.8e-38)) (<= M 1.55e-13)))))
     (* (/ c0 (* 2.0 w)) (* 2.0 (* (* (/ d D) (/ d D)) (/ (/ c0 w) h))))
     0.0)))
double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 4.1e-171) {
		tmp = 0.0;
	} else if ((M <= 2.05e-153) || !((M <= 1.85e-59) || (!(M <= 4.8e-38) && (M <= 1.55e-13)))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 4.1d-171) then
        tmp = 0.0d0
    else if ((m <= 2.05d-153) .or. (.not. (m <= 1.85d-59) .or. (.not. (m <= 4.8d-38)) .and. (m <= 1.55d-13))) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((d_1 / d) * (d_1 / d)) * ((c0 / w) / h)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 4.1e-171) {
		tmp = 0.0;
	} else if ((M <= 2.05e-153) || !((M <= 1.85e-59) || (!(M <= 4.8e-38) && (M <= 1.55e-13)))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 4.1e-171:
		tmp = 0.0
	elif (M <= 2.05e-153) or not ((M <= 1.85e-59) or (not (M <= 4.8e-38) and (M <= 1.55e-13))):
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)))
	else:
		tmp = 0.0
	return tmp

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 4.1e-171)
		tmp = 0.0;
	elseif ((M <= 2.05e-153) || !((M <= 1.85e-59) || (!(M <= 4.8e-38) && (M <= 1.55e-13))))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 / w) / h))));
	else
		tmp = 0.0;
	end
	return tmp
end

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 4.1e-171)
		tmp = 0.0;
	elseif ((M <= 2.05e-153) || ~(((M <= 1.85e-59) || (~((M <= 4.8e-38)) && (M <= 1.55e-13)))))
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 4.1e-171], 0.0, If[Or[LessEqual[M, 2.05e-153], N[Not[Or[LessEqual[M, 1.85e-59], And[N[Not[LessEqual[M, 4.8e-38]], $MachinePrecision], LessEqual[M, 1.55e-13]]]], $MachinePrecision]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;M \leq 4.1 \cdot 10^{-171}:\\
\;\;\;\;0\\

\mathbf{elif}\;M \leq 2.05 \cdot 10^{-153} \lor \neg \left(M \leq 1.85 \cdot 10^{-59} \lor \neg \left(M \leq 4.8 \cdot 10^{-38}\right) \land M \leq 1.55 \cdot 10^{-13}\right):\\
\;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\

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


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

Alternative 3: 39.1% accurate, 4.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M \leq 1.85 \cdot 10^{-177}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{-148}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{w \cdot \left(\frac{D}{d} \cdot \frac{h \cdot D}{d}\right)}\right)\\ \mathbf{elif}\;M \leq 4.6 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{-37} \lor \neg \left(M \leq 2.4 \cdot 10^{-13}\right):\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]
(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= M 1.85e-177)
     0.0
     (if (<= M 5.5e-148)
       (* t_0 (* 2.0 (/ c0 (* w (* (/ D d) (/ (* h D) d))))))
       (if (<= M 4.6e-65)
         0.0
         (if (or (<= M 4.2e-37) (not (<= M 2.4e-13)))
           (* t_0 (* 2.0 (* (* (/ d D) (/ d D)) (/ (/ c0 w) h))))
           0.0))))))
double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (M <= 1.85e-177) {
		tmp = 0.0;
	} else if (M <= 5.5e-148) {
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))));
	} else if (M <= 4.6e-65) {
		tmp = 0.0;
	} else if ((M <= 4.2e-37) || !(M <= 2.4e-13)) {
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if (m <= 1.85d-177) then
        tmp = 0.0d0
    else if (m <= 5.5d-148) then
        tmp = t_0 * (2.0d0 * (c0 / (w * ((d / d_1) * ((h * d) / d_1)))))
    else if (m <= 4.6d-65) then
        tmp = 0.0d0
    else if ((m <= 4.2d-37) .or. (.not. (m <= 2.4d-13))) then
        tmp = t_0 * (2.0d0 * (((d_1 / d) * (d_1 / d)) * ((c0 / w) / h)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (M <= 1.85e-177) {
		tmp = 0.0;
	} else if (M <= 5.5e-148) {
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))));
	} else if (M <= 4.6e-65) {
		tmp = 0.0;
	} else if ((M <= 4.2e-37) || !(M <= 2.4e-13)) {
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if M <= 1.85e-177:
		tmp = 0.0
	elif M <= 5.5e-148:
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))))
	elif M <= 4.6e-65:
		tmp = 0.0
	elif (M <= 4.2e-37) or not (M <= 2.4e-13):
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)))
	else:
		tmp = 0.0
	return tmp

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (M <= 1.85e-177)
		tmp = 0.0;
	elseif (M <= 5.5e-148)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(w * Float64(Float64(D / d) * Float64(Float64(h * D) / d))))));
	elseif (M <= 4.6e-65)
		tmp = 0.0;
	elseif ((M <= 4.2e-37) || !(M <= 2.4e-13))
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 / w) / h))));
	else
		tmp = 0.0;
	end
	return tmp
end

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if (M <= 1.85e-177)
		tmp = 0.0;
	elseif (M <= 5.5e-148)
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))));
	elseif (M <= 4.6e-65)
		tmp = 0.0;
	elseif ((M <= 4.2e-37) || ~((M <= 2.4e-13)))
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * ((c0 / w) / h)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 1.85e-177], 0.0, If[LessEqual[M, 5.5e-148], N[(t$95$0 * N[(2.0 * N[(c0 / N[(w * N[(N[(D / d), $MachinePrecision] * N[(N[(h * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 4.6e-65], 0.0, If[Or[LessEqual[M, 4.2e-37], N[Not[LessEqual[M, 2.4e-13]], $MachinePrecision]], N[(t$95$0 * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{c0}{2 \cdot w}\\
\mathbf{if}\;M \leq 1.85 \cdot 10^{-177}:\\
\;\;\;\;0\\

\mathbf{elif}\;M \leq 5.5 \cdot 10^{-148}:\\
\;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{w \cdot \left(\frac{D}{d} \cdot \frac{h \cdot D}{d}\right)}\right)\\

\mathbf{elif}\;M \leq 4.6 \cdot 10^{-65}:\\
\;\;\;\;0\\

\mathbf{elif}\;M \leq 4.2 \cdot 10^{-37} \lor \neg \left(M \leq 2.4 \cdot 10^{-13}\right):\\
\;\;\;\;t_0 \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\

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


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

Alternative 4: 47.9% accurate, 5.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := t_0 \cdot \left(2 \cdot \frac{c0}{\left(w \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{D}{d}\right)}\right)\\ \mathbf{if}\;c0 \leq -7.5 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq 2.15 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq 10^{-95}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{w \cdot \left(\frac{D}{d} \cdot \frac{h \cdot D}{d}\right)}\right)\\ \mathbf{elif}\;c0 \leq 6 \cdot 10^{-34}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]
(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* t_0 (* 2.0 (/ c0 (* (* w (/ D d)) (* h (/ D d))))))))
   (if (<= c0 -7.5e-109)
     t_1
     (if (<= c0 2.15e-159)
       0.0
       (if (<= c0 1e-95)
         (* t_0 (* 2.0 (/ c0 (* w (* (/ D d) (/ (* h D) d))))))
         (if (<= c0 6e-34) 0.0 t_1))))))
double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = t_0 * (2.0 * (c0 / ((w * (D / d)) * (h * (D / d)))));
	double tmp;
	if (c0 <= -7.5e-109) {
		tmp = t_1;
	} else if (c0 <= 2.15e-159) {
		tmp = 0.0;
	} else if (c0 <= 1e-95) {
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))));
	} else if (c0 <= 6e-34) {
		tmp = 0.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = t_0 * (2.0d0 * (c0 / ((w * (d / d_1)) * (h * (d / d_1)))))
    if (c0 <= (-7.5d-109)) then
        tmp = t_1
    else if (c0 <= 2.15d-159) then
        tmp = 0.0d0
    else if (c0 <= 1d-95) then
        tmp = t_0 * (2.0d0 * (c0 / (w * ((d / d_1) * ((h * d) / d_1)))))
    else if (c0 <= 6d-34) then
        tmp = 0.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = t_0 * (2.0 * (c0 / ((w * (D / d)) * (h * (D / d)))));
	double tmp;
	if (c0 <= -7.5e-109) {
		tmp = t_1;
	} else if (c0 <= 2.15e-159) {
		tmp = 0.0;
	} else if (c0 <= 1e-95) {
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))));
	} else if (c0 <= 6e-34) {
		tmp = 0.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = t_0 * (2.0 * (c0 / ((w * (D / d)) * (h * (D / d)))))
	tmp = 0
	if c0 <= -7.5e-109:
		tmp = t_1
	elif c0 <= 2.15e-159:
		tmp = 0.0
	elif c0 <= 1e-95:
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))))
	elif c0 <= 6e-34:
		tmp = 0.0
	else:
		tmp = t_1
	return tmp

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(Float64(w * Float64(D / d)) * Float64(h * Float64(D / d))))))
	tmp = 0.0
	if (c0 <= -7.5e-109)
		tmp = t_1;
	elseif (c0 <= 2.15e-159)
		tmp = 0.0;
	elseif (c0 <= 1e-95)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(w * Float64(Float64(D / d) * Float64(Float64(h * D) / d))))));
	elseif (c0 <= 6e-34)
		tmp = 0.0;
	else
		tmp = t_1;
	end
	return tmp
end

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = t_0 * (2.0 * (c0 / ((w * (D / d)) * (h * (D / d)))));
	tmp = 0.0;
	if (c0 <= -7.5e-109)
		tmp = t_1;
	elseif (c0 <= 2.15e-159)
		tmp = 0.0;
	elseif (c0 <= 1e-95)
		tmp = t_0 * (2.0 * (c0 / (w * ((D / d) * ((h * D) / d)))));
	elseif (c0 <= 6e-34)
		tmp = 0.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(2.0 * N[(c0 / N[(N[(w * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -7.5e-109], t$95$1, If[LessEqual[c0, 2.15e-159], 0.0, If[LessEqual[c0, 1e-95], N[(t$95$0 * N[(2.0 * N[(c0 / N[(w * N[(N[(D / d), $MachinePrecision] * N[(N[(h * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 6e-34], 0.0, t$95$1]]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{c0}{2 \cdot w}\\
t_1 := t_0 \cdot \left(2 \cdot \frac{c0}{\left(w \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{D}{d}\right)}\right)\\
\mathbf{if}\;c0 \leq -7.5 \cdot 10^{-109}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;c0 \leq 2.15 \cdot 10^{-159}:\\
\;\;\;\;0\\

\mathbf{elif}\;c0 \leq 10^{-95}:\\
\;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{w \cdot \left(\frac{D}{d} \cdot \frac{h \cdot D}{d}\right)}\right)\\

\mathbf{elif}\;c0 \leq 6 \cdot 10^{-34}:\\
\;\;\;\;0\\

\mathbf{else}:\\
\;\;\;\;t_1\\


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

Alternative 5: 33.7% accurate, 151.0× speedup?

\[\begin{array}{l} \\ 0 \end{array} \]
(FPCore (c0 w h D d M) :precision binary64 0.0)
double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

def code(c0, w, h, D, d, M):
	return 0.0

function code(c0, w, h, D, d, M)
	return 0.0
end

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

code[c0_, w_, h_, D_, d_, M_] := 0.0

\begin{array}{l}

\\
0
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023342 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))