Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 57.0%

Time: 14.8s

Alternatives: 11

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -1.192375562090518e+236
  2. w = 3.007395362163833e-234
  3. h = 4.9937336880044e-186
  4. D = 1.2854955046242029e-303
  5. d = -3.2348853885723915e+299
  6. M = -2.7643728041034343e-257

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: 57.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{d \cdot c0}{D \cdot w}\right)}^{2}}{h}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (pow (/ (* d c0) (* D w)) 2.0) h)
     0.0)))

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = pow(((d * c0) / (D * w)), 2.0) / h;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (((d * c0) / (D * w)) ^ 2.0) / h;
	else
		tmp = 0.0;
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[Power[N[(N[(d * c0), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 75.5%

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

   9. Applied rewrites 85.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 55.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{\frac{\frac{d \cdot c0}{h}}{w}}{D}, \frac{d}{D}, \frac{\frac{c0 \cdot d}{h \cdot w} \cdot \frac{d}{D}}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (*
      t_0
      (fma
       (/ (/ (/ (* d c0) h) w) D)
       (/ d D)
       (/ (* (/ (* c0 d) (* h w)) (/ d D)) D)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma(((((d * c0) / h) / w) / D), (d / D), ((((c0 * d) / (h * w)) * (d / D)) / D));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in c0 around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 83.2%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{\frac{d \cdot c0}{h}}{w}}{D}, \frac{d}{D}, \frac{\frac{c0 \cdot d}{h \cdot w} \cdot \frac{d}{D}}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot c0}{h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\frac{\frac{t\_0}{w}}{D}, \frac{d}{D}, \frac{t\_0}{\frac{D}{d} \cdot \left(D \cdot w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d c0) h))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (fma (/ (/ t_0 w) D) (/ d D) (/ t_0 (* (/ D d) (* D w)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * c0) / h;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(((t_0 / w) / D), (d / D), (t_0 / ((D / d) * (D * w))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * c0) / h)
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma(Float64(Float64(t_0 / w) / D), Float64(d / D), Float64(t_0 / Float64(Float64(D / d) * Float64(D * w)))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * c0), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(N[(t$95$0 / w), $MachinePrecision] / D), $MachinePrecision] * N[(d / D), $MachinePrecision] + N[(t$95$0 / N[(N[(D / d), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in c0 around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 82.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 55.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{\frac{\frac{d \cdot c0}{h}}{w}}{D}, \frac{d}{D}, \frac{\left(c0 \cdot d\right) \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (*
      t_0
      (fma
       (/ (/ (/ (* d c0) h) w) D)
       (/ d D)
       (/ (* (* c0 d) d) (* (* (* h w) D) D))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma(((((d * c0) / h) / w) / D), (d / D), (((c0 * d) * d) / (((h * w) * D) * D)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in c0 around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 81.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{\frac{d \cdot c0}{h}}{w}}{D}, \frac{d}{D}, \frac{\left(c0 \cdot d\right) \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{\frac{\frac{d \cdot c0}{h}}{w}}{D}, \frac{d}{D}, \left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (*
      t_0
      (fma
       (/ (/ (/ (* d c0) h) w) D)
       (/ d D)
       (* (* d c0) (/ d (* (* (* D D) w) h)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma(((((d * c0) / h) / w) / D), (d / D), ((d * c0) * (d / (((D * D) * w) * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in c0 around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 80.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{\left(\left(d \cdot d\right) \cdot c0\right) \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (/ (* (* (* d d) c0) 2.0) (* (* (* D D) h) w)))
     0.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot c0}{\left(\left(h \cdot w\right) \cdot w\right) \cdot D} \cdot \left(\frac{d}{D} \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (/ (* d c0) (* (* (* h w) w) D)) (* (/ d D) c0))
     0.0)))

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((d * c0) / (((h * w) * w) * D)) * ((d / D) * c0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((d * c0) / (((h * w) * w) * D)) * ((d / D) * c0);
	else
		tmp = 0.0;
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 75.5%

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

   9. Applied rewrites 73.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(\frac{d}{\left(w \cdot \left(D \cdot D\right)\right) \cdot h} \cdot \frac{d}{w}\right) \cdot \left(c0 \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (* (/ d (* (* w (* D D)) h)) (/ d w)) (* c0 c0))
     0.0)))

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((d / ((w * (D * D)) * h)) * (d / w)) * (c0 * c0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((d / ((w * (D * D)) * h)) * (d / w)) * (c0 * c0);
	else
		tmp = 0.0;
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(d / N[(N[(w * N[(D * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 69.2%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot d}{D \cdot \left(\left(\left(h \cdot w\right) \cdot D\right) \cdot w\right)} \cdot \left(c0 \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (/ (* d d) (* D (* (* (* h w) D) w))) (* c0 c0))
     0.0)))

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((d * d) / (D * (((h * w) * D) * w))) * (c0 * c0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((d * d) / (D * (((h * w) * D) * w))) * (c0 * c0);
	else
		tmp = 0.0;
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(d * d), $MachinePrecision] / N[(D * N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 60.5%

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

   9. Applied rewrites 67.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (/ (* (* d d) c0) (* (* (* D D) h) (* w w))) c0)
     0.0)))

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (((d * d) * c0) / (((D * D) * h) * (w * w))) * c0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (((d * d) * c0) / (((D * D) * h) * (w * w))) * c0;
	else
		tmp = 0.0;
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 77.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 65.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.9

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

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 34.2% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

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

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 = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.7%

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

3. Taylor expanded in c0 around -inf

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

   1. associate-/l* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. div0 N/A

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

   6. mul0-rgt N/A

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

   7. metadata-eval 33.9

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

5. Applied rewrites 33.9%

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

Reproduce

herbie shell --seed 2024307 
(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))))))