Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 43.2%

Time: 37.2s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. c0 = -2.116071654743034e+298
  2. w = 7.737491827856847e-209
  3. h = 3.320711322589492e-191
  4. D = 6.191181955389436e-216
  5. d = -2.680570770559146e+278
  6. M = 1.4202155276452658e+288

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: 24.4% 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: 43.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \frac{{\left(\mathsf{hypot}\left({\left({\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}\right)}^{0.25}, \frac{d}{D} \cdot \sqrt{\frac{c0}{h \cdot w}}\right)\right)}^{2}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h 2.5e-309)
   (*
    c0
    (/
     (pow
      (hypot
       (pow
        (- (pow (/ (* c0 (pow (/ d D) 2.0)) (* h w)) 2.0) (pow M 2.0))
        0.25)
       (* (/ d D) (sqrt (/ c0 (* h w)))))
      2.0)
     (* 2.0 w)))
   (pow
    (* (/ (* c0 (* d (* (sqrt 0.5) (sqrt 2.0)))) (* D w)) (sqrt (/ 1.0 h)))
    2.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 2.5e-309) {
		tmp = c0 * (pow(hypot(pow((pow(((c0 * pow((d / D), 2.0)) / (h * w)), 2.0) - pow(M, 2.0)), 0.25), ((d / D) * sqrt((c0 / (h * w))))), 2.0) / (2.0 * w));
	} else {
		tmp = pow((((c0 * (d * (sqrt(0.5) * sqrt(2.0)))) / (D * w)) * sqrt((1.0 / h))), 2.0);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 2.5e-309) {
		tmp = c0 * (Math.pow(Math.hypot(Math.pow((Math.pow(((c0 * Math.pow((d / D), 2.0)) / (h * w)), 2.0) - Math.pow(M, 2.0)), 0.25), ((d / D) * Math.sqrt((c0 / (h * w))))), 2.0) / (2.0 * w));
	} else {
		tmp = Math.pow((((c0 * (d * (Math.sqrt(0.5) * Math.sqrt(2.0)))) / (D * w)) * Math.sqrt((1.0 / h))), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= 2.5e-309:
		tmp = c0 * (math.pow(math.hypot(math.pow((math.pow(((c0 * math.pow((d / D), 2.0)) / (h * w)), 2.0) - math.pow(M, 2.0)), 0.25), ((d / D) * math.sqrt((c0 / (h * w))))), 2.0) / (2.0 * w))
	else:
		tmp = math.pow((((c0 * (d * (math.sqrt(0.5) * math.sqrt(2.0)))) / (D * w)) * math.sqrt((1.0 / h))), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= 2.5e-309)
		tmp = Float64(c0 * Float64((hypot((Float64((Float64(Float64(c0 * (Float64(d / D) ^ 2.0)) / Float64(h * w)) ^ 2.0) - (M ^ 2.0)) ^ 0.25), Float64(Float64(d / D) * sqrt(Float64(c0 / Float64(h * w))))) ^ 2.0) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(Float64(c0 * Float64(d * Float64(sqrt(0.5) * sqrt(2.0)))) / Float64(D * w)) * sqrt(Float64(1.0 / h))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= 2.5e-309)
		tmp = c0 * ((hypot((((((c0 * ((d / D) ^ 2.0)) / (h * w)) ^ 2.0) - (M ^ 2.0)) ^ 0.25), ((d / D) * sqrt((c0 / (h * w))))) ^ 2.0) / (2.0 * w));
	else
		tmp = (((c0 * (d * (sqrt(0.5) * sqrt(2.0)))) / (D * w)) * sqrt((1.0 / h))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, 2.5e-309], N[(c0 * N[(N[Power[N[Sqrt[N[Power[N[(N[Power[N[(N[(c0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision] ^ 2 + N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(c0 * N[(d * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 2.5000000000000022e-309

   1. Initial program 21.8%

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

   3. Simplified 40.6%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   6. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{{\left(\mathsf{hypot}\left({\left({\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}\right)}^{0.25}, \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)\right)}^{2}}}{2 \cdot w} \]

   if 2.5000000000000022e-309 < h

   1. Initial program 27.4%

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

   3. Simplified 26.6%

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

      1. add-sqr-sqrt 26.6%

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

      2. pow2 26.6%

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

   6. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}\right) \cdot \frac{c0}{w \cdot 2}}\right)}^{2}} \]

   7. Taylor expanded in c0 around inf 55.7%

      \[\leadsto {\color{blue}{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \frac{{\left(\mathsf{hypot}\left({\left({\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}\right)}^{0.25}, \frac{d}{D} \cdot \sqrt{\frac{c0}{h \cdot w}}\right)\right)}^{2}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 35.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{h \cdot w}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t\_0, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \frac{d}{D} \cdot \sqrt{t\_0}, \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* h w))))
   (if (<= h -9e-305)
     (*
      (/ c0 (* 2.0 w))
      (fma
       (sqrt (fma t_0 (pow (/ d D) 2.0) M))
       (* (/ d D) (sqrt t_0))
       (* (/ d D) (* (/ d D) (/ (/ c0 w) h)))))
     (pow
      (* (/ (* c0 (* d (* (sqrt 0.5) (sqrt 2.0)))) (* D w)) (sqrt (/ 1.0 h)))
      2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (h * w);
	double tmp;
	if (h <= -9e-305) {
		tmp = (c0 / (2.0 * w)) * fma(sqrt(fma(t_0, pow((d / D), 2.0), M)), ((d / D) * sqrt(t_0)), ((d / D) * ((d / D) * ((c0 / w) / h))));
	} else {
		tmp = pow((((c0 * (d * (sqrt(0.5) * sqrt(2.0)))) / (D * w)) * sqrt((1.0 / h))), 2.0);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(h * w))
	tmp = 0.0
	if (h <= -9e-305)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * fma(sqrt(fma(t_0, (Float64(d / D) ^ 2.0), M)), Float64(Float64(d / D) * sqrt(t_0)), Float64(Float64(d / D) * Float64(Float64(d / D) * Float64(Float64(c0 / w) / h)))));
	else
		tmp = Float64(Float64(Float64(c0 * Float64(d * Float64(sqrt(0.5) * sqrt(2.0)))) / Float64(D * w)) * sqrt(Float64(1.0 / h))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-305], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] + M), $MachinePrecision]], $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] + N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(c0 * N[(d * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.0000000000000003e-305

   1. Initial program 22.2%

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

   3. Simplified 23.2%

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

   6. Applied egg-rr 40.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - M}, \frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)} \]

   7. Taylor expanded in c0 around inf 25.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \color{blue}{\frac{d}{D} \cdot \sqrt{\frac{c0}{h \cdot w}}}, \frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right) \]
   8. Step-by-step derivation

      1. associate-*l/ 36.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. pow2 36.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. associate-*r* 38.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      4. associate-/r* 39.6%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\left(\color{blue}{\frac{\frac{c0}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   9. Applied egg-rr 26.0%

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

   if -9.0000000000000003e-305 < h

   1. Initial program 27.0%

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

   3. Simplified 26.3%

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

      1. add-sqr-sqrt 26.3%

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

      2. pow2 26.3%

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

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}\right) \cdot \frac{c0}{w \cdot 2}}\right)}^{2}} \]

   7. Taylor expanded in c0 around inf 54.9%

      \[\leadsto {\color{blue}{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{c0}{h \cdot w}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \frac{d}{D} \cdot \sqrt{\frac{c0}{h \cdot w}}, \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, t\_0, \sqrt{\mathsf{fma}\left(c0, t\_0, M\right) \cdot \left(c0 \cdot t\_0 - M\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D)))))))
   (if (<= h -9e-305)
     (*
      c0
      (/ (fma c0 t_0 (sqrt (* (fma c0 t_0 M) (- (* c0 t_0) M)))) (* 2.0 w)))
     (pow
      (* (/ (* c0 (* d (* (sqrt 0.5) (sqrt 2.0)))) (* D w)) (sqrt (/ 1.0 h)))
      2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double tmp;
	if (h <= -9e-305) {
		tmp = c0 * (fma(c0, t_0, sqrt((fma(c0, t_0, M) * ((c0 * t_0) - M)))) / (2.0 * w));
	} else {
		tmp = pow((((c0 * (d * (sqrt(0.5) * sqrt(2.0)))) / (D * w)) * sqrt((1.0 / h))), 2.0);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	tmp = 0.0
	if (h <= -9e-305)
		tmp = Float64(c0 * Float64(fma(c0, t_0, sqrt(Float64(fma(c0, t_0, M) * Float64(Float64(c0 * t_0) - M)))) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(Float64(c0 * Float64(d * Float64(sqrt(0.5) * sqrt(2.0)))) / Float64(D * w)) * sqrt(Float64(1.0 / h))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-305], N[(c0 * N[(N[(c0 * t$95$0 + N[Sqrt[N[(N[(c0 * t$95$0 + M), $MachinePrecision] * N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(c0 * N[(d * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.0000000000000003e-305

   1. Initial program 22.2%

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

   3. Simplified 41.3%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   if -9.0000000000000003e-305 < h

   1. Initial program 27.0%

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

   3. Simplified 26.3%

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

      1. add-sqr-sqrt 26.3%

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

      2. pow2 26.3%

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

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}\right) \cdot \frac{c0}{w \cdot 2}}\right)}^{2}} \]

   7. Taylor expanded in c0 around inf 54.9%

      \[\leadsto {\color{blue}{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot \left(d \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+232}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0}{D} \cdot \frac{d}{w \cdot \sqrt{h}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* h w) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 4e+232) t_1 (pow (* (/ c0 D) (/ d (* w (sqrt h)))) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((h * w) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 4e+232) {
		tmp = t_1;
	} else {
		tmp = pow(((c0 / D) * (d / (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c0 * (d_1 * d_1)) / ((h * w) * (d * d))
    t_1 = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    if (t_1 <= 4d+232) then
        tmp = t_1
    else
        tmp = ((c0 / d) * (d_1 / (w * sqrt(h)))) ** 2.0d0
    end if
    code = tmp
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)) / ((h * w) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 4e+232) {
		tmp = t_1;
	} else {
		tmp = Math.pow(((c0 / D) * (d / (w * Math.sqrt(h)))), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((h * w) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= 4e+232:
		tmp = t_1
	else:
		tmp = math.pow(((c0 / D) * (d / (w * math.sqrt(h)))), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(h * w) * Float64(D * D)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= 4e+232)
		tmp = t_1;
	else
		tmp = Float64(Float64(c0 / D) * Float64(d / Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((h * w) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= 4e+232)
		tmp = t_1;
	else
		tmp = ((c0 / D) * (d / (w * sqrt(h)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 4e+232], t$95$1, N[Power[N[(N[(c0 / D), $MachinePrecision] * N[(d / N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 4.00000000000000023e232

   1. Initial program 77.1%

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

   if 4.00000000000000023e232 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 14.3%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in c0 around inf 17.4%

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

      1. *-un-lft-identity 17.4%

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

      2. add-sqr-sqrt 15.2%

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

      3. pow2 15.2%

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

   7. Applied egg-rr 32.5%

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

      1. *-lft-identity 32.5%

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

      2. times-frac 31.8%

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

   9. Simplified 31.8%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 4 \cdot 10^{+232}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0}{D} \cdot \frac{d}{w \cdot \sqrt{h}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* h w) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 5e-145) t_1 (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((h * w) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 5e-145) {
		tmp = t_1;
	} else {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c0 * (d_1 * d_1)) / ((h * w) * (d * d))
    t_1 = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    if (t_1 <= 5d-145) then
        tmp = t_1
    else
        tmp = ((c0 * d_1) / (d * (w * sqrt(h)))) ** 2.0d0
    end if
    code = tmp
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)) / ((h * w) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 5e-145) {
		tmp = t_1;
	} else {
		tmp = Math.pow(((c0 * d) / (D * (w * Math.sqrt(h)))), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((h * w) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= 5e-145:
		tmp = t_1
	else:
		tmp = math.pow(((c0 * d) / (D * (w * math.sqrt(h)))), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(h * w) * Float64(D * D)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= 5e-145)
		tmp = t_1;
	else
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((h * w) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= 5e-145)
		tmp = t_1;
	else
		tmp = ((c0 * d) / (D * (w * sqrt(h)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 5e-145], t$95$1, N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 4.9999999999999998e-145

   1. Initial program 75.4%

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

   if 4.9999999999999998e-145 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 15.5%

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

   3. Simplified 16.5%

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

   5. Taylor expanded in c0 around inf 17.6%

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

      1. add-sqr-sqrt 15.5%

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

      2. pow2 15.5%

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

   7. Applied egg-rr 33.4%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, t\_0, \sqrt{\mathsf{fma}\left(c0, t\_0, M\right) \cdot \left(c0 \cdot t\_0 - M\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D)))))))
   (if (<= h -9e-305)
     (*
      c0
      (/ (fma c0 t_0 (sqrt (* (fma c0 t_0 M) (- (* c0 t_0) M)))) (* 2.0 w)))
     (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double tmp;
	if (h <= -9e-305) {
		tmp = c0 * (fma(c0, t_0, sqrt((fma(c0, t_0, M) * ((c0 * t_0) - M)))) / (2.0 * w));
	} else {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	tmp = 0.0
	if (h <= -9e-305)
		tmp = Float64(c0 * Float64(fma(c0, t_0, sqrt(Float64(fma(c0, t_0, M) * Float64(Float64(c0 * t_0) - M)))) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-305], N[(c0 * N[(N[(c0 * t$95$0 + N[Sqrt[N[(N[(c0 * t$95$0 + M), $MachinePrecision] * N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.0000000000000003e-305

   1. Initial program 22.2%

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

   3. Simplified 41.3%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   if -9.0000000000000003e-305 < h

   1. Initial program 27.0%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in c0 around inf 24.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. pow2 24.6%

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

   7. Applied egg-rr 52.9%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, t\_0, \sqrt{\left(c0 \cdot t\_0 - M\right) \cdot \left(M + \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}\right)\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D)))))))
   (if (<= h -9e-305)
     (*
      c0
      (/
       (fma
        c0
        t_0
        (sqrt
         (* (- (* c0 t_0) M) (+ M (* (/ d D) (* (/ d D) (/ (/ c0 w) h)))))))
       (* 2.0 w)))
     (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double tmp;
	if (h <= -9e-305) {
		tmp = c0 * (fma(c0, t_0, sqrt((((c0 * t_0) - M) * (M + ((d / D) * ((d / D) * ((c0 / w) / h))))))) / (2.0 * w));
	} else {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	tmp = 0.0
	if (h <= -9e-305)
		tmp = Float64(c0 * Float64(fma(c0, t_0, sqrt(Float64(Float64(Float64(c0 * t_0) - M) * Float64(M + Float64(Float64(d / D) * Float64(Float64(d / D) * Float64(Float64(c0 / w) / h))))))) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-305], N[(c0 * N[(N[(c0 * t$95$0 + N[Sqrt[N[(N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision] * N[(M + N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.0000000000000003e-305

   1. Initial program 22.2%

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

   3. Simplified 41.3%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 41.3%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) + M\right)} \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. associate-*r/ 36.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(c0 \cdot \color{blue}{\frac{d \cdot d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. *-commutative 36.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(w \cdot \left(h \cdot D\right)\right) \cdot D}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      4. associate-*r* 34.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right)} \cdot D} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      5. associate-*r* 27.5%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      6. associate-/l* 27.5%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      7. frac-times 26.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      8. associate-*l/ 26.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\frac{c0 \cdot \frac{d \cdot d}{D \cdot D}}{w \cdot h}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      9. times-frac 36.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\frac{c0 \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}}{w \cdot h} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      10. pow2 36.1%

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\frac{c0 \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}}}{w \cdot h} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   6. Applied egg-rr 36.1%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} + M\right)} \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]
   7. Step-by-step derivation

      1. associate-*l/ 36.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. pow2 36.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. associate-*r* 38.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      4. associate-/r* 39.6%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\left(\color{blue}{\frac{\frac{c0}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   8. Applied egg-rr 39.6%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(\color{blue}{\left(\frac{\frac{c0}{w}}{h} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}} + M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   if -9.0000000000000003e-305 < h

   1. Initial program 27.0%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in c0 around inf 24.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. pow2 24.6%

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

   7. Applied egg-rr 52.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right) \cdot \left(M + \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}\right)\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, t\_0, \sqrt{M \cdot \left(c0 \cdot t\_0 - M\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D)))))))
   (if (<= h -9e-305)
     (* c0 (/ (fma c0 t_0 (sqrt (* M (- (* c0 t_0) M)))) (* 2.0 w)))
     (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double tmp;
	if (h <= -9e-305) {
		tmp = c0 * (fma(c0, t_0, sqrt((M * ((c0 * t_0) - M)))) / (2.0 * w));
	} else {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	tmp = 0.0
	if (h <= -9e-305)
		tmp = Float64(c0 * Float64(fma(c0, t_0, sqrt(Float64(M * Float64(Float64(c0 * t_0) - M)))) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -9e-305], N[(c0 * N[(N[(c0 * t$95$0 + N[Sqrt[N[(M * N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.0000000000000003e-305

   1. Initial program 22.2%

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

   3. Simplified 41.3%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around 0 22.4%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{M} \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   if -9.0000000000000003e-305 < h

   1. Initial program 27.0%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in c0 around inf 24.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. pow2 24.6%

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

   7. Applied egg-rr 52.9%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{M \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{{D}^{2}} \cdot \frac{{d}^{2}}{h \cdot w}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -2.3e-307)
   (* c0 (/ (* 2.0 (* (/ c0 (pow D 2.0)) (/ (pow d 2.0) (* h w)))) (* 2.0 w)))
   (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2.3e-307) {
		tmp = c0 * ((2.0 * ((c0 / pow(D, 2.0)) * (pow(d, 2.0) / (h * w)))) / (2.0 * w));
	} else {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

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) :: tmp
    if (h <= (-2.3d-307)) then
        tmp = c0 * ((2.0d0 * ((c0 / (d ** 2.0d0)) * ((d_1 ** 2.0d0) / (h * w)))) / (2.0d0 * w))
    else
        tmp = ((c0 * d_1) / (d * (w * sqrt(h)))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2.3e-307) {
		tmp = c0 * ((2.0 * ((c0 / Math.pow(D, 2.0)) * (Math.pow(d, 2.0) / (h * w)))) / (2.0 * w));
	} else {
		tmp = Math.pow(((c0 * d) / (D * (w * Math.sqrt(h)))), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -2.3e-307:
		tmp = c0 * ((2.0 * ((c0 / math.pow(D, 2.0)) * (math.pow(d, 2.0) / (h * w)))) / (2.0 * w))
	else:
		tmp = math.pow(((c0 * d) / (D * (w * math.sqrt(h)))), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -2.3e-307)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 / (D ^ 2.0)) * Float64((d ^ 2.0) / Float64(h * w)))) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -2.3e-307)
		tmp = c0 * ((2.0 * ((c0 / (D ^ 2.0)) * ((d ^ 2.0) / (h * w)))) / (2.0 * w));
	else
		tmp = ((c0 * d) / (D * (w * sqrt(h)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -2.3e-307], N[(c0 * N[(N[(2.0 * N[(N[(c0 / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -2.2999999999999999e-307

   1. Initial program 22.0%

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

   3. Simplified 40.9%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around inf 28.6%

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

      1. times-frac 28.6%

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

   7. Simplified 28.6%

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

   if -2.2999999999999999e-307 < h

   1. Initial program 27.2%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in c0 around inf 24.8%

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

      1. add-sqr-sqrt 24.8%

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

      2. pow2 24.8%

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

   7. Applied egg-rr 53.3%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{{D}^{2}} \cdot \frac{{d}^{2}}{h \cdot w}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot {d}^{2}}{\left(h \cdot w\right) \cdot {D}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -2.3e-307)
   (* c0 (/ (* 2.0 (/ (* c0 (pow d 2.0)) (* (* h w) (pow D 2.0)))) (* 2.0 w)))
   (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2.3e-307) {
		tmp = c0 * ((2.0 * ((c0 * pow(d, 2.0)) / ((h * w) * pow(D, 2.0)))) / (2.0 * w));
	} else {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	}
	return tmp;
}

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) :: tmp
    if (h <= (-2.3d-307)) then
        tmp = c0 * ((2.0d0 * ((c0 * (d_1 ** 2.0d0)) / ((h * w) * (d ** 2.0d0)))) / (2.0d0 * w))
    else
        tmp = ((c0 * d_1) / (d * (w * sqrt(h)))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2.3e-307) {
		tmp = c0 * ((2.0 * ((c0 * Math.pow(d, 2.0)) / ((h * w) * Math.pow(D, 2.0)))) / (2.0 * w));
	} else {
		tmp = Math.pow(((c0 * d) / (D * (w * Math.sqrt(h)))), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -2.3e-307:
		tmp = c0 * ((2.0 * ((c0 * math.pow(d, 2.0)) / ((h * w) * math.pow(D, 2.0)))) / (2.0 * w))
	else:
		tmp = math.pow(((c0 * d) / (D * (w * math.sqrt(h)))), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -2.3e-307)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(h * w) * (D ^ 2.0)))) / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -2.3e-307)
		tmp = c0 * ((2.0 * ((c0 * (d ^ 2.0)) / ((h * w) * (D ^ 2.0)))) / (2.0 * w));
	else
		tmp = ((c0 * d) / (D * (w * sqrt(h)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -2.3e-307], N[(c0 * N[(N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -2.2999999999999999e-307

   1. Initial program 22.0%

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

   3. Simplified 40.9%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around inf 28.6%

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

   if -2.2999999999999999e-307 < h

   1. Initial program 27.2%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in c0 around inf 24.8%

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

      1. add-sqr-sqrt 24.8%

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

      2. pow2 24.8%

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

   7. Applied egg-rr 53.3%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot {d}^{2}}{\left(h \cdot w\right) \cdot {D}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{h \cdot w}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* h w))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (*
    (/ c0 (* 2.0 w))
    (+ (sqrt (- (* t_1 t_1) (* M M))) (* t_0 (* (/ d D) (/ d D)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (h * w);
	double t_1 = t_0 * ((d * d) / (D * D));
	return (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
}

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
    real(8) :: t_1
    t_0 = c0 / (h * w)
    t_1 = t_0 * ((d_1 * d_1) / (d * d))
    code = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (t_0 * ((d_1 / d) * (d_1 / d))))
end function

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (h * w)
	t_1 = t_0 * ((d * d) / (D * D))
	return (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(h * w))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = c0 / (h * w);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
end

Wolfram

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

Derivation

1. Initial program 24.8%

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

3. Simplified 24.9%

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

   1. times-frac 24.5%

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

6. Applied egg-rr 24.5%

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

7. Final simplification 24.5%

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

Alternative 12: 24.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{h \cdot w}\\ t_1 := t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* h w))) (t_1 (* t_0 (* (/ d D) (/ d D)))))
   (*
    (/ c0 (* 2.0 w))
    (+ (* t_0 (/ (* d d) (* D D))) (sqrt (- (* t_1 t_1) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (h * w);
	double t_1 = t_0 * ((d / D) * (d / D));
	return (c0 / (2.0 * w)) * ((t_0 * ((d * d) / (D * D))) + sqrt(((t_1 * t_1) - (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
    real(8) :: t_1
    t_0 = c0 / (h * w)
    t_1 = t_0 * ((d_1 / d) * (d_1 / d))
    code = (c0 / (2.0d0 * w)) * ((t_0 * ((d_1 * d_1) / (d * d))) + sqrt(((t_1 * t_1) - (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 / (h * w);
	double t_1 = t_0 * ((d / D) * (d / D));
	return (c0 / (2.0 * w)) * ((t_0 * ((d * d) / (D * D))) + Math.sqrt(((t_1 * t_1) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (h * w)
	t_1 = t_0 * ((d / D) * (d / D))
	return (c0 / (2.0 * w)) * ((t_0 * ((d * d) / (D * D))) + math.sqrt(((t_1 * t_1) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(h * w))
	t_1 = Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(t_0 * Float64(Float64(d * d) / Float64(D * D))) + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = c0 / (h * w);
	t_1 = t_0 * ((d / D) * (d / D));
	tmp = (c0 / (2.0 * w)) * ((t_0 * ((d * d) / (D * D))) + sqrt(((t_1 * t_1) - (M * M))));
end

Wolfram

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

Derivation

1. Initial program 24.8%

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

3. Simplified 24.9%

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

   1. times-frac 24.5%

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

6. Applied egg-rr 24.5%

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

   1. times-frac 24.5%

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

8. Applied egg-rr 24.7%

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

9. Final simplification 24.7%

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

Alternative 13: 24.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* h w)) (/ (* d d) (* 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 / (h * w)) * ((d * d) / (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 / (h * w)) * ((d_1 * d_1) / (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 / (h * w)) * ((d * d) / (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 / (h * w)) * ((d * d) / (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(h * w)) * Float64(Float64(d * d) / 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 / (h * w)) * ((d * d) / (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[(h * w), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $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]]

Derivation

1. Initial program 24.8%

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

3. Simplified 24.9%

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

5. Final simplification 24.9%

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

Alternative 14: 0.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{M \cdot \sqrt{-1}}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* c0 (/ (* M (sqrt -1.0)) (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * ((M * sqrt(-1.0)) / (2.0 * w));
}

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
    code = c0 * ((m * sqrt((-1.0d0))) / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * ((M * Math.sqrt(-1.0)) / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * ((M * math.sqrt(-1.0)) / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(Float64(M * sqrt(-1.0)) / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * ((M * sqrt(-1.0)) / (2.0 * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(N[(M * N[Sqrt[-1.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

3. Simplified 40.2%

   \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w}} \]
4. Add Preprocessing

5. Taylor expanded in c0 around 0 0.0%

   \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

6. Final simplification 0.0%

   \[\leadsto c0 \cdot \frac{M \cdot \sqrt{-1}}{2 \cdot w} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))))))