Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 45.9%

Time: 15.7s

Alternatives: 3

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = 1.4865621803859512e+213
  2. w = -4.338262246573144e-284
  3. h = 2.397078459140097e-281
  4. D = 8.1657034875581315e-289
  5. d = -6.732752625590586e+271
  6. M = -5.912546069459659e+91

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(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)))))))

C

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))));
}

Fortran

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

Java

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))));
}

Python

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))))

Julia

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

MATLAB

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

Wolfram

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]]

Initial Program: 24.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(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)))))))

C

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))));
}

Fortran

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

Java

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))));
}

Python

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))))

Julia

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

MATLAB

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

Wolfram

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]]

Alternative 1: 45.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot c0}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot w\right) \cdot D} \cdot \left(d \cdot c0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 1.45e-198)
   0.0
   (* (/ (* d c0) (* (* (* (* h w) D) w) D)) (* d c0))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.45e-198) {
		tmp = 0.0;
	} else {
		tmp = ((d * c0) / ((((h * w) * D) * w) * D)) * (d * c0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 1.45d-198) then
        tmp = 0.0d0
    else
        tmp = ((d_1 * c0) / ((((h * w) * d) * w) * d)) * (d_1 * c0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.45e-198) {
		tmp = 0.0;
	} else {
		tmp = ((d * c0) / ((((h * w) * D) * w) * D)) * (d * c0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 1.45e-198:
		tmp = 0.0
	else:
		tmp = ((d * c0) / ((((h * w) * D) * w) * D)) * (d * c0)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 1.45e-198)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d * c0) / Float64(Float64(Float64(Float64(h * w) * D) * w) * D)) * Float64(d * c0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 1.45e-198)
		tmp = 0.0;
	else
		tmp = ((d * c0) / ((((h * w) * D) * w) * D)) * (d * c0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 1.45e-198], 0.0, N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.45e-198

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \frac{-1}{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \frac{-1}{2}}{w}} \]

      3. distribute-lft1-in N/A

         \[\leadsto \frac{\left({c0}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \cdot \frac{-1}{2}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\left({c0}^{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \frac{-1}{2}}{w} \]

      5. mul0-lft N/A

         \[\leadsto \frac{\left({c0}^{2} \cdot \color{blue}{0}\right) \cdot \frac{-1}{2}}{w} \]

      6. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \frac{-1}{2}}{w} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{w} \]

      8. div0 33.0

         \[\leadsto \color{blue}{0} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{0} \]

   if 1.45e-198 < M

   1. Initial program 27.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \cdot {d}^{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot c0}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot c0}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2} \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot c0}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \cdot {d}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot c0}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \cdot {d}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot c0}{\color{blue}{\left({D}^{2} \cdot h\right)} \cdot {w}^{2}} \cdot {d}^{2} \]

      9. unpow2 N/A

         \[\leadsto \frac{c0 \cdot c0}{\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot {w}^{2}} \cdot {d}^{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot {w}^{2}} \cdot {d}^{2} \]

      11. unpow2 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \cdot {d}^{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \cdot {d}^{2} \]

      13. unpow2 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \color{blue}{\left(d \cdot d\right)} \]

      14. lower-*.f64 31.6

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \color{blue}{\left(d \cdot d\right)} \]

   5. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \left(d \cdot d\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto \left(c0 \cdot \left(\frac{c0}{\left(w \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w} \cdot d\right)\right) \cdot \color{blue}{d} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.3%

      \[\leadsto \frac{d \cdot c0}{\left(w \cdot \left(\left(h \cdot w\right) \cdot D\right)\right) \cdot D} \cdot \color{blue}{\left(d \cdot c0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot c0}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot w\right) \cdot D} \cdot \left(d \cdot c0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot w\right) \cdot D} \cdot c0\right) \cdot c0\right) \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 1.45e-198)
   0.0
   (* (* (* (/ d (* (* (* (* h w) D) w) D)) c0) c0) d)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.45e-198) {
		tmp = 0.0;
	} else {
		tmp = (((d / ((((h * w) * D) * w) * D)) * c0) * c0) * d;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 1.45d-198) then
        tmp = 0.0d0
    else
        tmp = (((d_1 / ((((h * w) * d) * w) * d)) * c0) * c0) * d_1
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.45e-198) {
		tmp = 0.0;
	} else {
		tmp = (((d / ((((h * w) * D) * w) * D)) * c0) * c0) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 1.45e-198:
		tmp = 0.0
	else:
		tmp = (((d / ((((h * w) * D) * w) * D)) * c0) * c0) * d
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 1.45e-198)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(d / Float64(Float64(Float64(Float64(h * w) * D) * w) * D)) * c0) * c0) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 1.45e-198)
		tmp = 0.0;
	else
		tmp = (((d / ((((h * w) * D) * w) * D)) * c0) * c0) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 1.45e-198], 0.0, N[(N[(N[(N[(d / N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.45e-198

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \frac{-1}{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \frac{-1}{2}}{w}} \]

      3. distribute-lft1-in N/A

         \[\leadsto \frac{\left({c0}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \cdot \frac{-1}{2}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\left({c0}^{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \frac{-1}{2}}{w} \]

      5. mul0-lft N/A

         \[\leadsto \frac{\left({c0}^{2} \cdot \color{blue}{0}\right) \cdot \frac{-1}{2}}{w} \]

      6. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \frac{-1}{2}}{w} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{w} \]

      8. div0 33.0

         \[\leadsto \color{blue}{0} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{0} \]

   if 1.45e-198 < M

   1. Initial program 27.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \cdot {d}^{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot c0}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot c0}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot {d}^{2} \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot c0}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \cdot {d}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot c0}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \cdot {d}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot c0}{\color{blue}{\left({D}^{2} \cdot h\right)} \cdot {w}^{2}} \cdot {d}^{2} \]

      9. unpow2 N/A

         \[\leadsto \frac{c0 \cdot c0}{\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot {w}^{2}} \cdot {d}^{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot {w}^{2}} \cdot {d}^{2} \]

      11. unpow2 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \cdot {d}^{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \cdot {d}^{2} \]

      13. unpow2 N/A

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \color{blue}{\left(d \cdot d\right)} \]

      14. lower-*.f64 31.6

          \[\leadsto \frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \color{blue}{\left(d \cdot d\right)} \]

   5. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \left(d \cdot d\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto \left(c0 \cdot \left(\frac{c0}{\left(w \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w} \cdot d\right)\right) \cdot \color{blue}{d} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.9%

      \[\leadsto \left(c0 \cdot \left(c0 \cdot \frac{d}{\left(w \cdot \left(\left(h \cdot w\right) \cdot D\right)\right) \cdot D}\right)\right) \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot w\right) \cdot D} \cdot c0\right) \cdot c0\right) \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 33.8% accurate, 156.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 0.0)

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = 0.0d0
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := 0.0

Derivation

1. Initial program 25.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \frac{-1}{2}} \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \frac{-1}{2}}{w}} \]

   3. distribute-lft1-in N/A

      \[\leadsto \frac{\left({c0}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \cdot \frac{-1}{2}}{w} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\left({c0}^{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \frac{-1}{2}}{w} \]

   5. mul0-lft N/A

      \[\leadsto \frac{\left({c0}^{2} \cdot \color{blue}{0}\right) \cdot \frac{-1}{2}}{w} \]

   6. mul0-rgt N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \frac{-1}{2}}{w} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{w} \]

   8. div0 27.9

      \[\leadsto \color{blue}{0} \]

5. Applied rewrites 27.9%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))))