Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.2% → 54.4%

Time: 10.4s

Alternatives: 11

Speedup: 3.7×

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: 24.2% 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: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{\frac{\left(2 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{d}}{d} \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (/ (/ (* (* 2.0 c0) (* d d)) (* (* h w) D)) D))
     (* (/ (/ (* (* (* M D) (* M D)) h) d) d) -0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((((2.0 * c0) * (d * d)) / ((h * w) * D)) / D);
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((((2.0 * c0) * (d * d)) / ((h * w) * D)) / D);
	} else {
		tmp = (((((M * D) * (M * D)) * h) / d) / d) * -0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * ((((2.0 * c0) * (d * d)) / ((h * w) * D)) / D)
	else:
		tmp = (((((M * D) * (M * D)) * h) / d) / d) * -0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(2.0 * c0) * Float64(d * d)) / Float64(Float64(h * w) * D)) / D));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / d) / d) * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((((2.0 * c0) * (d * d)) / ((h * w) * D)) / D);
	else
		tmp = (((((M * D) * (M * D)) * h) / d) / d) * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(N[(N[(2.0 * c0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $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 73.6%

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

   3. Taylor expanded in c0 around inf

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

      1. 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 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. unpow-prod-down N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

   10. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 44.5

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

   12. Applied rewrites 44.5%

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

Alternative 2: 54.4% accurate, 0.7× speedup

Math

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

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)) (* (/ 2.0 (* (* h w) D)) (/ (* (* d d) c0) D)))
     (* (/ (/ (* (* (* M D) (* M D)) h) 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)) * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D));
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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)) * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D));
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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)) * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D))
	else:
		tmp = (((((M * D) * (M * D)) * h) / 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(2.0 / Float64(Float64(h * w) * D)) * Float64(Float64(Float64(d * d) * c0) / D)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / 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)) * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D));
	else
		tmp = (((((M * D) * (M * D)) * h) / 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[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $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 73.6%

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

   3. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 0.0

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\right) \]

   5. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

         \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

      3. lower-+.f64 0.0

         \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   8. Taylor expanded in D around 0

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

   10. Applied rewrites 54.9%

       \[\leadsto \frac{c0}{w + w} \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{h} \cdot \frac{d \cdot d}{w}, 2, \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w\right) \cdot {D}^{4}}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right)}{D \cdot D}} \]

   11. Taylor expanded in c0 around inf

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

       1. associate-*r/ N/A

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

       2. pow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. associate-*l* N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-*r* N/A

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

       15. pow2 N/A

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

       16. lower-/.f64 N/A

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

   13. Applied rewrites 77.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. unpow-prod-down N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

   10. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 44.5

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

   12. Applied rewrites 44.5%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{d}}{d} \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.4% accurate, 0.7× speedup

Math

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

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)) (/ (* 2.0 (* (* d c0) d)) (* (* (* h w) D) D)))
     (* (/ (/ (* (* (* M D) (* M D)) h) 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)) * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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)) * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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)) * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D))
	else:
		tmp = (((((M * D) * (M * D)) * h) / 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(2.0 * Float64(Float64(d * c0) * d)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / 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)) * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	else
		tmp = (((((M * D) * (M * D)) * h) / 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[(2.0 * N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $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 73.6%

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

   3. Taylor expanded in c0 around inf

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

      1. 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 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   6. 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. *-commutative N/A

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

      9. lower-*.f64 75.7

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 75.7

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

   9. Applied rewrites 75.7%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. unpow-prod-down N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

   10. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 44.5

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

   12. Applied rewrites 44.5%

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

Alternative 4: 53.2% accurate, 0.7× speedup

Math

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

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)
     (/ (/ (* (* d c0) (* d c0)) (* (* D w) (* D w))) h)
     (* (/ (/ (* (* (* M D) (* M D)) h) 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 = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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 = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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 = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h
	else:
		tmp = (((((M * D) * (M * D)) * h) / 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(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(D * w) * Float64(D * w))) / h);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / 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 = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	else
		tmp = (((((M * D) * (M * D)) * h) / 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[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $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 73.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

   7. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 72.9

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

   9. Applied rewrites 72.9%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 72.9

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

   11. Applied rewrites 72.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. unpow-prod-down N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

   10. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 44.5

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

   12. Applied rewrites 44.5%

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

Alternative 5: 50.4% accurate, 0.7× speedup

Math

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

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)
     (/ (/ (* (* d c0) (* d c0)) (* D D)) (* (* w w) h))
     (* (/ (/ (* (* (* M D) (* M D)) h) 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 = (((d * c0) * (d * c0)) / (D * D)) / ((w * w) * h);
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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 = (((d * c0) * (d * c0)) / (D * D)) / ((w * w) * h);
	} else {
		tmp = (((((M * D) * (M * D)) * h) / 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 = (((d * c0) * (d * c0)) / (D * D)) / ((w * w) * h)
	else:
		tmp = (((((M * D) * (M * D)) * h) / 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(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(D * D)) / Float64(Float64(w * w) * h));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / 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 = (((d * c0) * (d * c0)) / (D * D)) / ((w * w) * h);
	else
		tmp = (((((M * D) * (M * D)) * h) / 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[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $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 73.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. unpow-prod-down N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

   10. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 44.5

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

   12. Applied rewrites 44.5%

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

Alternative 6: 47.5% accurate, 0.7× speedup

Math

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

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 D) (* M D)) h) 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 * D) * (M * D)) * h) / 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 * D) * (M * D)) * h) / 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 * D) * (M * D)) * h) / 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(M * D) * Float64(M * D)) * h) / 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 * D) * (M * D)) * h) / 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[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $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 73.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{{w}^{2} \cdot h} \]

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-*.f64 N/A

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

      25. pow2 N/A

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

   7. Applied rewrites 53.8%

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

   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. Add Preprocessing

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. unpow-prod-down N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 44.5

          \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]

   10. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot h}{d}}{d} \cdot -0.25 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 44.5

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

   12. Applied rewrites 44.5%

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

Alternative 7: 45.7% accurate, 0.7× speedup

Math

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

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 D) (* M D)) h) (* 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 * D) * (M * D)) * h) / (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 * D) * (M * D)) * h) / (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 * D) * (M * D)) * h) / (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(M * D) * Float64(M * D)) * h) / Float64(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 * D) * (M * D)) * h) / (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[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $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 73.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{{w}^{2} \cdot h} \]

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-*.f64 N/A

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

      25. pow2 N/A

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

   7. Applied rewrites 53.8%

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

   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. Add Preprocessing

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 42.5

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 42.5%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 42.5

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

   10. Applied rewrites 42.5%

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

Alternative 8: 30.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{d \cdot d}\\ \mathbf{if}\;M \leq 2.3 \cdot 10^{+156}:\\ \;\;\;\;\left(D \cdot \left(D \cdot \left(t\_0 \cdot \left(M \cdot M\right)\right)\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot \left(M \cdot \left(M \cdot t\_0\right)\right)\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ h (* d d))))
   (if (<= M 2.3e+156)
     (* (* D (* D (* t_0 (* M M)))) -0.25)
     (* (* (* D D) (* M (* M t_0))) -0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h / (d * d);
	double tmp;
	if (M <= 2.3e+156) {
		tmp = (D * (D * (t_0 * (M * M)))) * -0.25;
	} else {
		tmp = ((D * D) * (M * (M * t_0))) * -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) :: t_0
    real(8) :: tmp
    t_0 = h / (d_1 * d_1)
    if (m <= 2.3d+156) then
        tmp = (d * (d * (t_0 * (m * m)))) * (-0.25d0)
    else
        tmp = ((d * d) * (m * (m * t_0))) * (-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 t_0 = h / (d * d);
	double tmp;
	if (M <= 2.3e+156) {
		tmp = (D * (D * (t_0 * (M * M)))) * -0.25;
	} else {
		tmp = ((D * D) * (M * (M * t_0))) * -0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h / (d * d)
	tmp = 0
	if M <= 2.3e+156:
		tmp = (D * (D * (t_0 * (M * M)))) * -0.25
	else:
		tmp = ((D * D) * (M * (M * t_0))) * -0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h / Float64(d * d))
	tmp = 0.0
	if (M <= 2.3e+156)
		tmp = Float64(Float64(D * Float64(D * Float64(t_0 * Float64(M * M)))) * -0.25);
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(M * Float64(M * t_0))) * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h / (d * d);
	tmp = 0.0;
	if (M <= 2.3e+156)
		tmp = (D * (D * (t_0 * (M * M)))) * -0.25;
	else
		tmp = ((D * D) * (M * (M * t_0))) * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2.3e+156], N[(N[(D * N[(D * N[(t$95$0 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * N[(M * N[(M * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2999999999999999e156

   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. Add Preprocessing

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 33.4

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 33.4%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*r* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 29.7

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

   10. Applied rewrites 29.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. pow2 N/A

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

       11. associate-*r/ N/A

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

       12. lower-*.f64 N/A

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

       13. associate-*r/ N/A

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

       14. pow2 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. pow2 N/A

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

       20. lift-*.f64 32.0

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

   12. Applied rewrites 32.0%

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

   if 2.2999999999999999e156 < M

   1. Initial program 0.0%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 14.0

          \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

   8. Applied rewrites 14.0%

      \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*r* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 0.0

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

   10. Applied rewrites 0.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 14.7

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

   12. Applied rewrites 14.7%

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

Alternative 9: 31.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in D around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 15.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

6. Taylor expanded in c0 around 0

   \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
7. 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. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 31.6

       \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

8. Applied rewrites 31.6%

   \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 31.6

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

10. Applied rewrites 31.6%

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

Alternative 10: 29.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (D * (D * ((h / (d * d)) * (M * M)))) * -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 = (d * (d * ((h / (d_1 * d_1)) * (m * m)))) * (-0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in D around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 15.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

6. Taylor expanded in c0 around 0

   \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
7. 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. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 31.6

       \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot -0.25 \]

8. Applied rewrites 31.6%

   \[\leadsto \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d} \cdot \color{blue}{-0.25} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-prod-down N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 27.0

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

10. Applied rewrites 27.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 N/A

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

    9. lift-/.f64 N/A

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

    10. pow2 N/A

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

    11. associate-*r/ N/A

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

    12. lower-*.f64 N/A

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

    13. associate-*r/ N/A

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

    14. pow2 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 N/A

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

    17. lift-/.f64 N/A

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

    18. lift-*.f64 N/A

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

    19. pow2 N/A

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

    20. lift-*.f64 29.2

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

12. Applied rewrites 29.2%

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

Alternative 11: 0.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right) \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (sqrt(-1.0) * 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
    code = (c0 / (w + w)) * (sqrt((-1.0d0)) * m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in c0 around 0

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 0.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   3. lower-+.f64 0.0

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

7. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025073 
(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))))))