Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.4% → 68.2%

Time: 10.5s

Alternatives: 11

Speedup: 77.7×

• could not determine a ground truth

  1. c0 = -1.8007703777139465e+298
  2. w = -2.962564929394273e-273
  3. h = -3.6037364111729084e-292
  4. D = 1.3650635920040177e-257
  5. d = -1.5829860075061084e+306
  6. M = 4.673470561462938e-140

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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: 25.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left(\left(\left(c0 \cdot d\right) \cdot \frac{d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\\ \end{array} \end{array} \]

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)
     (* (/ c0 (+ w w)) (* (* (* c0 d) (/ d (* (* (* h w) D) D))) 2.0))
     (* (/ (/ (* (* (* (* M M) h) D) D) d) d) 0.25))))

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 = (c0 / (w + w)) * (((c0 * d) * (d / (((h * w) * D) * D))) * 2.0);
	} else {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	}
	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 = (c0 / (w + w)) * (((c0 * d) * (d / (((h * w) * D) * D))) * 2.0);
	} else {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	}
	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 = (c0 / (w + w)) * (((c0 * d) * (d / (((h * w) * D) * D))) * 2.0)
	else:
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25
	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(c0 / Float64(w + w)) * Float64(Float64(Float64(c0 * d) * Float64(d / Float64(Float64(Float64(h * w) * D) * D))) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) * D) / d) / d) * 0.25);
	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 = (c0 / (w + w)) * (((c0 * d) * (d / (((h * w) * D) * D))) * 2.0);
	else
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	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[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * d), $MachinePrecision] * N[(d / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]]]

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 75.9%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

   8. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

   10. Applied rewrites 79.8%

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

   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. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 42.6

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 42.6%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left(\left(\left(\frac{d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \cdot d\right) \cdot c0\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\\ \end{array} \end{array} \]

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)
     (* (/ c0 (+ w w)) (* (* (* (/ d (* (* (* h w) D) D)) d) c0) 2.0))
     (* (/ (/ (* (* (* (* M M) h) D) D) d) d) 0.25))))

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 = (c0 / (w + w)) * ((((d / (((h * w) * D) * D)) * d) * c0) * 2.0);
	} else {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	}
	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 = (c0 / (w + w)) * ((((d / (((h * w) * D) * D)) * d) * c0) * 2.0);
	} else {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	}
	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 = (c0 / (w + w)) * ((((d / (((h * w) * D) * D)) * d) * c0) * 2.0)
	else:
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25
	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(c0 / Float64(w + w)) * Float64(Float64(Float64(Float64(d / Float64(Float64(Float64(h * w) * D) * D)) * d) * c0) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) * D) / d) / d) * 0.25);
	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 = (c0 / (w + w)) * ((((d / (((h * w) * D) * D)) * d) * c0) * 2.0);
	else
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	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[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(d / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]]]

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 75.9%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

   8. Applied rewrites 75.7%

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

   10. Applied rewrites 77.5%

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

   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. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 42.6

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 42.6%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\\ \end{array} \end{array} \]

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)
     (* (* c0 c0) (/ (* d d) (* (* (* D D) h) (* w w))))
     (* (/ (/ (* (* (* (* M M) h) D) D) d) d) 0.25))))

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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)));
	} else {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	}
	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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)));
	} else {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	}
	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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)))
	else:
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25
	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(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(Float64(D * D) * h) * Float64(w * w))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) * D) / d) / d) * 0.25);
	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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)));
	else
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	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[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]]]

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 75.9%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   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. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 42.6

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 42.6%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 62.4%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 48.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right) \cdot 0.25}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.5e+36)
   (/ (* (* (* (* (* M M) h) D) (/ D d)) 0.25) d)
   (* (/ (/ (* (* (* M (* M h)) D) D) d) d) 0.25)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.5e+36) {
		tmp = (((((M * M) * h) * D) * (D / d)) * 0.25) / d;
	} else {
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 1.5d+36) then
        tmp = (((((m * m) * h) * d) * (d / d_1)) * 0.25d0) / d_1
    else
        tmp = (((((m * (m * h)) * d) * d) / d_1) / d_1) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.5e+36) {
		tmp = (((((M * M) * h) * D) * (D / d)) * 0.25) / d;
	} else {
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.5e+36:
		tmp = (((((M * M) * h) * D) * (D / d)) * 0.25) / d
	else:
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.5e+36)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) * Float64(D / d)) * 0.25) / d);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * Float64(M * h)) * D) * D) / d) / d) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.5e+36)
		tmp = (((((M * M) * h) * D) * (D / d)) * 0.25) / d;
	else
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.5e+36], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.5e36

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 33.1

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 33.1%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 46.8%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d} \cdot \frac{1}{4}}{d} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d} \cdot \frac{1}{4}}{d} \]

   11. Applied rewrites 45.8%

       \[\leadsto \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right) \cdot 0.25}{d} \]

   if 1.5e36 < d

   1. Initial program 26.0%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 34.9

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 34.9%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       5. lower-*.f64 50.4

          \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]

   11. Applied rewrites 50.4%

       \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 47.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h 5e+166)
   (* (/ (/ (* (* (* (* M M) h) D) D) d) d) 0.25)
   (* (/ (/ (* (* (* M M) (* D h)) D) d) d) 0.25)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 5e+166) {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	} else {
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (h <= 5d+166) then
        tmp = ((((((m * m) * h) * d) * d) / d_1) / d_1) * 0.25d0
    else
        tmp = (((((m * m) * (d * h)) * d) / d_1) / d_1) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 5e+166) {
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	} else {
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= 5e+166:
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25
	else:
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= 5e+166)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) * D) / d) / d) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * Float64(D * h)) * D) / d) / d) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= 5e+166)
		tmp = ((((((M * M) * h) * D) * D) / d) / d) * 0.25;
	else
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, 5e+166], N[(N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.0000000000000002e166

   1. Initial program 25.6%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 34.0

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 34.0%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 48.1%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]

   if 5.0000000000000002e166 < h

   1. Initial program 22.8%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 4.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 29.9

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 46.8%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(h \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       8. pow2 N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       10. lift-*.f64 44.7

           \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \]

   11. Applied rewrites 44.7%

       \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 47.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h 5e+166)
   (* (/ (/ (* (* (* M (* M h)) D) D) d) d) 0.25)
   (* (/ (/ (* (* (* M M) (* D h)) D) d) d) 0.25)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 5e+166) {
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25;
	} else {
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (h <= 5d+166) then
        tmp = (((((m * (m * h)) * d) * d) / d_1) / d_1) * 0.25d0
    else
        tmp = (((((m * m) * (d * h)) * d) / d_1) / d_1) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 5e+166) {
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25;
	} else {
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= 5e+166:
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25
	else:
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= 5e+166)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * Float64(M * h)) * D) * D) / d) / d) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * Float64(D * h)) * D) / d) / d) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= 5e+166)
		tmp = (((((M * (M * h)) * D) * D) / d) / d) * 0.25;
	else
		tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, 5e+166], N[(N[(N[(N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.0000000000000002e166

   1. Initial program 25.6%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 34.0

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 34.0%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 48.1%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       5. lower-*.f64 48.4

          \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]

   11. Applied rewrites 48.4%

       \[\leadsto \frac{\frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]

   if 5.0000000000000002e166 < h

   1. Initial program 22.8%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 4.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 29.9

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   9. Applied rewrites 46.8%

      \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(h \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       8. pow2 N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

       10. lift-*.f64 44.7

           \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \]

   11. Applied rewrites 44.7%

       \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 45.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.25 \cdot 10^{+148}:\\ \;\;\;\;\frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot 0.25}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.25e+148) (/ (* (* (* (* (* M M) h) D) D) 0.25) (* d d)) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.25e+148) {
		tmp = (((((M * M) * h) * D) * D) * 0.25) / (d * d);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 1.25d+148) then
        tmp = (((((m * m) * h) * d) * d) * 0.25d0) / (d_1 * d_1)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.25e+148) {
		tmp = (((((M * M) * h) * D) * D) * 0.25) / (d * d);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.25e+148:
		tmp = (((((M * M) * h) * D) * D) * 0.25) / (d * d)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.25e+148)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) * D) * 0.25) / Float64(d * d));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.25e+148)
		tmp = (((((M * M) * h) * D) * D) * 0.25) / (d * d);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.25e+148], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * 0.25), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if d < 1.25000000000000006e148

   1. Initial program 26.2%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 35.0

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 35.0%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      4. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}}{{d}^{\color{blue}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}}{{d}^{\color{blue}{2}}} \]

   9. Applied rewrites 41.4%

      \[\leadsto \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot 0.25}{d \cdot \color{blue}{d}} \]

   if 1.25000000000000006e148 < d

   1. Initial program 22.5%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 35.1

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in c0 around -inf

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

   7. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 38.0%

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

Alternative 8: 40.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (/ (* (* (* M M) (* D h)) D) d) d) 0.25))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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 = (((((m * m) * (d * h)) * d) / d_1) / d_1) * 0.25d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
}

Python

def code(c0, w, h, D, d, M):
	return (((((M * M) * (D * h)) * D) / d) / d) * 0.25

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(Float64(Float64(Float64(M * M) * Float64(D * h)) * D) / d) / d) * 0.25)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (((((M * M) * (D * h)) * D) / d) / d) * 0.25;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]

Derivation

1. Initial program 25.4%

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

2. Taylor expanded in c0 around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 3.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

5. Taylor expanded in c0 around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

   6. pow2 N/A

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

   9. pow2 N/A

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

   11. pow2 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

   12. lift-*.f64 33.7

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

7. Applied rewrites 33.7%

   \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d}}{d} \cdot \frac{1}{4} \]

9. Applied rewrites 48.0%

   \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot 0.25 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    4. pow2 N/A

       \[\leadsto \frac{\frac{\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    5. associate-*l* N/A

       \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(h \cdot D\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\left({M}^{2} \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    8. pow2 N/A

       \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot \frac{1}{4} \]

    10. lift-*.f64 45.7

        \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \]

11. Applied rewrites 45.7%

    \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot h\right)\right) \cdot D}{d}}{d} \cdot 0.25 \]
12. Add Preprocessing

Alternative 9: 35.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{+110}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D \cdot D}{d \cdot d}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.6e+110) (* (* (* (* M M) h) (/ (* D D) (* d d))) 0.25) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.6e+110) {
		tmp = (((M * M) * h) * ((D * D) / (d * d))) * 0.25;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 1.6d+110) then
        tmp = (((m * m) * h) * ((d * d) / (d_1 * d_1))) * 0.25d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.6e+110) {
		tmp = (((M * M) * h) * ((D * D) / (d * d))) * 0.25;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.6e+110:
		tmp = (((M * M) * h) * ((D * D) / (d * d))) * 0.25
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.6e+110)
		tmp = Float64(Float64(Float64(Float64(M * M) * h) * Float64(Float64(D * D) / Float64(d * d))) * 0.25);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.6e+110)
		tmp = (((M * M) * h) * ((D * D) / (d * d))) * 0.25;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.6e+110], N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if d < 1.59999999999999997e110

   1. Initial program 26.1%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 34.5

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 34.5%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{d \cdot d} \cdot \frac{1}{4} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      10. pow2 N/A

          \[\leadsto \left(\left({M}^{2} \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left({M}^{2} \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      12. pow2 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      16. pow2 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D \cdot D}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D \cdot D}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      18. pow2 N/A

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D \cdot D}{d \cdot d}\right) \cdot \frac{1}{4} \]

      19. lift-*.f64 34.0

          \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D \cdot D}{d \cdot d}\right) \cdot 0.25 \]

   9. Applied rewrites 34.0%

      \[\leadsto \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D \cdot D}{d \cdot d}\right) \cdot 0.25 \]

   if 1.59999999999999997e110 < d

   1. Initial program 23.5%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in c0 around -inf

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

   7. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 37.9%

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

Alternative 10: 34.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 3.6 \cdot 10^{-153}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 3.6e-153) 0.0 (* (* (* D D) (* (* M M) (/ h (* d d)))) 0.25)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3.6e-153) {
		tmp = 0.0;
	} else {
		tmp = ((D * D) * ((M * M) * (h / (d * d)))) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 3.6d-153) then
        tmp = 0.0d0
    else
        tmp = ((d * d) * ((m * m) * (h / (d_1 * d_1)))) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3.6e-153) {
		tmp = 0.0;
	} else {
		tmp = ((D * D) * ((M * M) * (h / (d * d)))) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 3.6e-153:
		tmp = 0.0
	else:
		tmp = ((D * D) * ((M * M) * (h / (d * d)))) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 3.6e-153)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(Float64(M * M) * Float64(h / Float64(d * d)))) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 3.6e-153)
		tmp = 0.0;
	else
		tmp = ((D * D) * ((M * M) * (h / (d * d)))) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 3.6e-153], 0.0, N[(N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.5999999999999998e-153

   1. Initial program 24.5%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 37.4

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

   4. Applied rewrites 37.4%

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

   5. Taylor expanded in c0 around -inf

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

   7. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 35.2%

       \[\leadsto 0 \]

   if 3.5999999999999998e-153 < D

   1. Initial program 27.5%

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

   2. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 4.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{w}, -0.5, \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25\right) \cdot \left(c0 \cdot c0\right)} \]

   5. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      12. lift-*.f64 35.8

          \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

   7. Applied rewrites 35.8%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{d \cdot d} \cdot \frac{1}{4} \]

      8. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d} \cdot \frac{1}{4} \]

      9. pow2 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot d} \cdot \frac{1}{4} \]

      10. pow2 N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

      11. associate-/l* N/A

          \[\leadsto \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      12. lower-*.f64 N/A

          \[\leadsto \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      13. pow2 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot \frac{1}{4} \]

      15. associate-/l* N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)\right) \cdot \frac{1}{4} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)\right) \cdot \frac{1}{4} \]

      17. pow2 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right)\right) \cdot \frac{1}{4} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right)\right) \cdot \frac{1}{4} \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right)\right) \cdot \frac{1}{4} \]

      20. pow2 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right) \cdot \frac{1}{4} \]

      21. lift-*.f64 34.0

          \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right) \cdot 0.25 \]

   9. Applied rewrites 34.0%

      \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right) \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 33.4% accurate, 77.7× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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 25.4%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 37.9

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

4. Applied rewrites 37.9%

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

5. Taylor expanded in c0 around -inf

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

7. Applied rewrites 24.7%

   \[\leadsto \color{blue}{\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5} \]

8. Taylor expanded in c0 around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 33.4%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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))))))