Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 44.7%

Time: 10.0s

Alternatives: 12

Speedup: 2.2×

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: 24.9% 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: 44.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := c0 \cdot t\_0\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, t\_0, M\right)\right)}^{0.5}, {\left(t\_2 - M\right)}^{0.5}, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d (* (* h w) D)) (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* c0 t_0))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_1 (fma (pow (fma c0 t_0 M) 0.5) (pow (- t_2 M) 0.5) t_2))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / ((h * w) * D)) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = c0 * t_0;
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(pow(fma(c0, t_0, M), 0.5), pow((t_2 - M), 0.5), t_2);
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(Float64(h * w) * D)) * Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(c0 * t_0)
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma((fma(c0, t_0, M) ^ 0.5), (Float64(t_2 - M) ^ 0.5), t_2));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[Power[N[(c0 * t$95$0 + M), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(t$95$2 - M), $MachinePrecision], 0.5], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 32.6

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

   5. Applied rewrites 32.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 32.9

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

   7. Applied rewrites 32.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 39.0

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

   9. Applied rewrites 39.0%

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

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 44.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, {\left(t\_0 - M\right)}^{0.5}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* (/ d (* (* h w) D)) (/ d D))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (fma (/ (* d (sqrt (/ c0 (* h w)))) D) (pow (- t_0 M) 0.5) t_0))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * ((d / ((h * w) * D)) * (d / D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(((d * sqrt((c0 / (h * w)))) / D), pow((t_0 - M), 0.5), t_0);
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(Float64(d / Float64(Float64(h * w) * D)) * Float64(d / D)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma(Float64(Float64(d * sqrt(Float64(c0 / Float64(h * w)))) / D), (Float64(t_0 - M) ^ 0.5), t_0));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(N[(d / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(N[(d * N[Sqrt[N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] * N[Power[N[(t$95$0 - M), $MachinePrecision], 0.5], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 32.6

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

   5. Applied rewrites 32.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 32.9

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

   7. Applied rewrites 32.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 39.0

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

   9. Applied rewrites 39.0%

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

   10. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot \sqrt{\frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot \sqrt{\frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot \sqrt{\frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       5. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot \sqrt{\frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot \sqrt{\frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       7. lift-*.f64 18.4

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

   12. Applied rewrites 18.4%

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

   13. Taylor expanded in D around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{\frac{1}{2}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

       5. lift-*.f64 20.9

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

   15. Applied rewrites 20.9%

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

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 44.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\left(d \cdot d\right) \cdot \left(t\_0 + {t\_0}^{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* (* D D) (* h w))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (* (* d d) (+ t_0 (pow t_0 1.0))))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / ((D * D) * (h * w));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((d * d) * (t_0 + pow(t_0, 1.0)));
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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) * (h * w));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((d * d) * (t_0 + Math.pow(t_0, 1.0)));
	} else {
		tmp = ((Math.pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / ((D * D) * (h * w))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * ((d * d) * (t_0 + math.pow(t_0, 1.0)))
	else:
		tmp = ((math.pow(-(M * M), 0.5) * c0) / w) * 0.5
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(Float64(D * D) * Float64(h * w)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(d * d) * Float64(t_0 + (t_0 ^ 1.0))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / ((D * D) * (h * w));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * ((d * d) * (t_0 + (t_0 ^ 1.0)));
	else
		tmp = (((-(M * M) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(d * d), $MachinePrecision] * N[(t$95$0 + N[Power[t$95$0, 1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

   6. Applied rewrites 33.2%

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

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 42.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(t\_0 + {t\_0}^{1}\right)\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* (* D D) (* h w))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* 0.5 (/ (* c0 (* (* d d) (+ t_0 (pow t_0 1.0)))) w))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / ((D * D) * (h * w));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = 0.5 * ((c0 * ((d * d) * (t_0 + pow(t_0, 1.0)))) / w);
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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) * (h * w));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * ((c0 * ((d * d) * (t_0 + Math.pow(t_0, 1.0)))) / w);
	} else {
		tmp = ((Math.pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / ((D * D) * (h * w))
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = 0.5 * ((c0 * ((d * d) * (t_0 + math.pow(t_0, 1.0)))) / w)
	else:
		tmp = ((math.pow(-(M * M), 0.5) * c0) / w) * 0.5
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(Float64(D * D) * Float64(h * w)))
	t_1 = 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_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(0.5 * Float64(Float64(c0 * Float64(Float64(d * d) * Float64(t_0 + (t_0 ^ 1.0)))) / w));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / ((D * D) * (h * w));
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = 0.5 * ((c0 * ((d * d) * (t_0 + (t_0 ^ 1.0)))) / w);
	else
		tmp = (((-(M * M) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(0.5 * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] * N[(t$95$0 + N[Power[t$95$0, 1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 33.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} + {\left(\frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right)}^{1}\right)\right)}{w}} \]

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 39.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \mathsf{fma}\left(\frac{c0}{h}, \frac{d \cdot d}{w}, {\left(\frac{t\_0}{h \cdot w}\right)}^{1}\right)}{\left(D \cdot D\right) \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (*
      0.5
      (/
       (* c0 (fma (/ c0 h) (/ (* d d) w) (pow (/ t_0 (* h w)) 1.0)))
       (* (* D D) w)))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = 0.5 * ((c0 * fma((c0 / h), ((d * d) / w), pow((t_0 / (h * w)), 1.0))) / ((D * D) * w));
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(0.5 * Float64(Float64(c0 * fma(Float64(c0 / h), Float64(Float64(d * d) / w), (Float64(t_0 / Float64(h * w)) ^ 1.0))) / Float64(Float64(D * D) * w)));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(0.5 * N[(N[(c0 * N[(N[(c0 / h), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / w), $MachinePrecision] + N[Power[N[(t$95$0 / N[(h * w), $MachinePrecision]), $MachinePrecision], 1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in D around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 31.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \mathsf{fma}\left(\frac{c0}{h}, \frac{d \cdot d}{w}, {\left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}\right)}^{1}\right)}{\left(D \cdot D\right) \cdot w}} \]

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 38.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \sqrt{M \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

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 (+ t_1 (sqrt (* M (- (* c0 (/ (* d d) (* (* (* h w) D) D))) M)))))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

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 * (t_1 + sqrt((M * ((c0 * ((d * d) / (((h * w) * D) * D))) - M))));
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 * (t_1 + Math.sqrt((M * ((c0 * ((d * d) / (((h * w) * D) * D))) - M))));
	} else {
		tmp = ((Math.pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 * (t_1 + math.sqrt((M * ((c0 * ((d * d) / (((h * w) * D) * D))) - M))))
	else:
		tmp = ((math.pow(-(M * M), 0.5) * c0) / w) * 0.5
	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(t_1 + sqrt(Float64(M * Float64(Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(h * w) * D) * D))) - M)))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	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 * (t_1 + sqrt((M * ((c0 * ((d * d) / (((h * w) * D) * D))) - M))));
	else
		tmp = (((-(M * M) ^ 0.5) * c0) / w) * 0.5;
	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[(t$95$1 + N[Sqrt[N[(M * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.4%

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

   4. Taylor expanded in c0 around 0

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

   6. Applied rewrites 14.7%

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

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 36.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot {\left(c0 \cdot \sqrt{\frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(c0 \cdot \left(h \cdot w\right)\right)}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

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 (pow (* c0 (sqrt (/ (* d d) (* (* D D) (* c0 (* h w)))))) 2.0))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

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 * pow((c0 * sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))), 2.0);
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 * Math.pow((c0 * Math.sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))), 2.0);
	} else {
		tmp = ((Math.pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 * math.pow((c0 * math.sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))), 2.0)
	else:
		tmp = ((math.pow(-(M * M), 0.5) * c0) / w) * 0.5
	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(c0 * sqrt(Float64(Float64(d * d) / Float64(Float64(D * D) * Float64(c0 * Float64(h * w)))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	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 * ((c0 * sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))) ^ 2.0);
	else
		tmp = (((-(M * M) ^ 0.5) * c0) / w) * 0.5;
	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[Power[N[(c0 * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(c0 * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in c0 around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

   6. Applied rewrites 25.6%

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

   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 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 33.0% accurate, 0.7× speedup

Math

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

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)
     (* 0.5 (/ (* c0 (/ (* c0 (* (* d d) h)) (* (* D D) w))) (* (* h h) w)))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

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 = 0.5 * ((c0 * ((c0 * ((d * d) * h)) / ((D * D) * w))) / ((h * h) * w));
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 = 0.5 * ((c0 * ((c0 * ((d * d) * h)) / ((D * D) * w))) / ((h * h) * w));
	} else {
		tmp = ((Math.pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 = 0.5 * ((c0 * ((c0 * ((d * d) * h)) / ((D * D) * w))) / ((h * h) * w))
	else:
		tmp = ((math.pow(-(M * M), 0.5) * c0) / w) * 0.5
	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(0.5 * Float64(Float64(c0 * Float64(Float64(c0 * Float64(Float64(d * d) * h)) / Float64(Float64(D * D) * w))) / Float64(Float64(h * h) * w)));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	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 = 0.5 * ((c0 * ((c0 * ((d * d) * h)) / ((D * D) * w))) / ((h * h) * w));
	else
		tmp = (((-(M * M) ^ 0.5) * c0) / w) * 0.5;
	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[(0.5 * N[(N[(c0 * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(h * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow1 22.2

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}}{\left(h \cdot \color{blue}{h}\right) \cdot w} \]

   8. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}}{\left(h \cdot \color{blue}{h}\right) \cdot w} \]

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 32.7% accurate, 0.7× speedup

Math

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

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)
     (* 0.5 (/ (* (* c0 c0) (* d d)) (* (* D D) (* h (* w w)))))
     (* (/ (* (pow (- (* M M)) 0.5) c0) w) 0.5))))

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 = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))));
	} else {
		tmp = ((pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))));
	} else {
		tmp = ((Math.pow(-(M * M), 0.5) * c0) / w) * 0.5;
	}
	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 = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))))
	else:
		tmp = ((math.pow(-(M * M), 0.5) * c0) / w) * 0.5
	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(0.5 * Float64(Float64(Float64(c0 * c0) * Float64(d * d)) / Float64(Float64(D * D) * Float64(h * Float64(w * w)))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M * M)) ^ 0.5) * c0) / w) * 0.5);
	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 = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))));
	else
		tmp = (((-(M * M) ^ 0.5) * c0) / w) * 0.5;
	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[(0.5 * N[(N[(N[(c0 * c0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $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 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]

   7. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 25.1

          \[\leadsto 0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

   9. Applied rewrites 25.1%

      \[\leadsto 0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

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

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.7

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.7%

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 31.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 0:\\ \;\;\;\;\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 0.0)
   (* (/ (* (sqrt (- (* M M))) c0) w) 0.5)
   (* 0.5 (/ (* (* c0 c0) (* d d)) (* (* D D) (* h (* w w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 0.0) {
		tmp = ((sqrt(-(M * M)) * c0) / w) * 0.5;
	} else {
		tmp = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 0.0d0) then
        tmp = ((sqrt(-(m * m)) * c0) / w) * 0.5d0
    else
        tmp = 0.5d0 * (((c0 * c0) * (d_1 * d_1)) / ((d * d) * (h * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 0.0) {
		tmp = ((Math.sqrt(-(M * M)) * c0) / w) * 0.5;
	} else {
		tmp = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 0.0:
		tmp = ((math.sqrt(-(M * M)) * c0) / w) * 0.5
	else:
		tmp = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 0.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(-Float64(M * M))) * c0) / w) * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 * c0) * Float64(d * d)) / Float64(Float64(D * D) * Float64(h * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 0.0)
		tmp = ((sqrt(-(M * M)) * c0) / w) * 0.5;
	else
		tmp = 0.5 * (((c0 * c0) * (d * d)) / ((D * D) * (h * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 0.0], N[(N[(N[(N[Sqrt[(-N[(M * M), $MachinePrecision])], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[(N[(c0 * c0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 0.0

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 15.1

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 15.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]

   if 0.0 < (*.f64 M M)

   1. Initial program 24.9%

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]

   7. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 25.1

          \[\leadsto 0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

   9. Applied rewrites 25.1%

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

Alternative 11: 15.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (* (sqrt (- (* M M))) c0) w) 0.5))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((sqrt(-(M * M)) * c0) / w) * 0.5;
}

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 = ((sqrt(-(m * m)) * c0) / w) * 0.5d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((Math.sqrt(-(M * M)) * c0) / w) * 0.5;
}

Python

def code(c0, w, h, D, d, M):
	return ((math.sqrt(-(M * M)) * c0) / w) * 0.5

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(sqrt(Float64(-Float64(M * M))) * c0) / w) * 0.5)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((sqrt(-(M * M)) * c0) / w) * 0.5;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[Sqrt[(-N[(M * M), $MachinePrecision])], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 24.9%

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

2. Taylor expanded in c0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 15.1

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

4. Applied rewrites 15.1%

   \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
5. Add Preprocessing

Alternative 12: 0.0% accurate, 5.2× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * 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 / (2.0d0 * 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 / (2.0 * w)) * (Math.sqrt(-1.0) * M);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. Taylor expanded in M around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
3. 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

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

4. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025137 
(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))))))