Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 98.2%

Time: 5.3s

Alternatives: 12

Speedup: 1.1×

• 52.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ \mathbf{if}\;\left(t\_0 - \frac{t\_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(t\_0 - 0.25 \cdot {\left(w \cdot r\right)}^{2}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \frac{t\_1 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))) (t_1 (* 0.125 (- 3.0 (* 2.0 v)))))
   (if (<=
        (- (- t_0 (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)
        (- INFINITY))
     (- (- t_0 (* 0.25 (pow (* w r) 2.0))) 4.5)
     (- (- t_0 (/ (* t_1 (* (* r w) (* r w))) (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double tmp;
	if (((t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -((double) INFINITY)) {
		tmp = (t_0 - (0.25 * pow((w * r), 2.0))) - 4.5;
	} else {
		tmp = (t_0 - ((t_1 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double tmp;
	if (((t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - (0.25 * Math.pow((w * r), 2.0))) - 4.5;
	} else {
		tmp = (t_0 - ((t_1 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = 0.125 * (3.0 - (2.0 * v))
	tmp = 0
	if ((t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -math.inf:
		tmp = (t_0 - (0.25 * math.pow((w * r), 2.0))) - 4.5
	else:
		tmp = (t_0 - ((t_1 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	tmp = 0.0
	if (Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(0.25 * (Float64(w * r) ^ 2.0))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(t_1 * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = 0.125 * (3.0 - (2.0 * v));
	tmp = 0.0;
	if (((t_0 - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -Inf)
		tmp = (t_0 - (0.25 * ((w * r) ^ 2.0))) - 4.5;
	else
		tmp = (t_0 - ((t_1 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 - N[(N[(t$95$1 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-Infinity)], N[(N[(t$95$0 - N[(0.25 * N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(t$95$1 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 81.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      3. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot {\left(w \cdot r\right)}^{\color{blue}{2}}\right) - \frac{9}{2} \]

      4. lower-pow.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{4} \cdot {\left(w \cdot r\right)}^{\color{blue}{2}}\right) - \frac{9}{2} \]

      5. lower-*.f64 98.1

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot {\left(w \cdot r\right)}^{2}\right) - 4.5 \]

   5. Applied rewrites 98.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 88.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 99.7

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 99.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+219} \lor \neg \left(t\_0 \leq -1.5\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (or (<= t_0 -1e+219) (not (<= t_0 -1.5)))
     (/ (/ (fma (* -1.5 r) r 2.0) r) r)
     -1.5)))

C

double code(double v, double w, double r) {
	double t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if ((t_0 <= -1e+219) || !(t_0 <= -1.5)) {
		tmp = (fma((-1.5 * r), r, 2.0) / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if ((t_0 <= -1e+219) || !(t_0 <= -1.5))
		tmp = Float64(Float64(fma(Float64(-1.5 * r), r, 2.0) / r) / r);
	else
		tmp = -1.5;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+219], N[Not[LessEqual[t$95$0, -1.5]], $MachinePrecision]], N[(N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], -1.5]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -9.99999999999999965e218 or -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 56.8

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{\color{blue}{r \cdot r}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      12. lift-*.f64 77.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \]

   7. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      2. pow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r}}{r} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-3}{2} \cdot r, r, 2\right)}{r}}{r} \]

      8. lower-*.f64 77.9

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \]

   9. Applied rewrites 77.9%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \]

   if -9.99999999999999965e218 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 90.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 45.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+219} \lor \neg \left(\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+219}:\\ \;\;\;\;\frac{\frac{\left(-1.5 \cdot r\right) \cdot r}{r}}{r}\\ \mathbf{elif}\;t\_0 \leq -1.5:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_0 -1e+219)
     (/ (/ (* (* -1.5 r) r) r) r)
     (if (<= t_0 -1.5) -1.5 (/ (fma (* -1.5 r) r 2.0) (* r r))))))

C

double code(double v, double w, double r) {
	double t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_0 <= -1e+219) {
		tmp = (((-1.5 * r) * r) / r) / r;
	} else if (t_0 <= -1.5) {
		tmp = -1.5;
	} else {
		tmp = fma((-1.5 * r), r, 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_0 <= -1e+219)
		tmp = Float64(Float64(Float64(Float64(-1.5 * r) * r) / r) / r);
	elseif (t_0 <= -1.5)
		tmp = -1.5;
	else
		tmp = Float64(fma(Float64(-1.5 * r), r, 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+219], N[(N[(N[(N[(-1.5 * r), $MachinePrecision] * r), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[t$95$0, -1.5], -1.5, N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -9.99999999999999965e218

   1. Initial program 82.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 6.3

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{\color{blue}{r \cdot r}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      12. lift-*.f64 52.1

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \]

   7. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}} \]

   8. Taylor expanded in r around inf

      \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2}}{r}}{r} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2}}{r}}{r} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2}}{r}}{r} \]

      3. pow2 N/A

         \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot \frac{-3}{2}}{r}}{r} \]

      4. lift-*.f64 47.3

         \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r} \]

   10. Applied rewrites 47.3%

       \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot \frac{-3}{2}}{r}}{r} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot \frac{-3}{2}}{r}}{r} \]

       3. pow2 N/A

          \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2}}{r}}{r} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2}}{r}}{r} \]

       5. pow2 N/A

          \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right)}{r}}{r} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(\frac{-3}{2} \cdot r\right) \cdot r}{r}}{r} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{-3}{2} \cdot r\right) \cdot r}{r}}{r} \]

       8. lower-*.f64 47.3

          \[\leadsto \frac{\frac{\left(-1.5 \cdot r\right) \cdot r}{r}}{r} \]

   12. Applied rewrites 47.3%

       \[\leadsto \frac{\frac{\left(-1.5 \cdot r\right) \cdot r}{r}}{r} \]

   if -9.99999999999999965e218 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 90.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 45.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

      \[\leadsto -1.5 \]

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2} \cdot r, r, 2\right)}{\color{blue}{r} \cdot r} \]

      5. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r} \]

   7. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{\color{blue}{r} \cdot r} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(0.125 \cdot \left(3 - \left(v + v\right)\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -1.5)
   (-
    (- 3.0 (/ (* (* 0.125 (- 3.0 (+ v v))) (* (* r w) (* r w))) (- 1.0 v)))
    4.5)
   (/ (/ (fma (* -1.5 r) r 2.0) r) r)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - (((0.125 * (3.0 - (v + v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (fma((-1.5 * r), r, 2.0) / r) / r;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v + v))) * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(fma(Float64(-1.5 * r), r, 2.0) / r) / r);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(N[(N[(0.125 * N[(3.0 - N[(v + v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 85.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 92.6

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 92.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.6%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{2 \cdot v}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      2. count-2-rev N/A

         \[\leadsto \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{\left(v + v\right)}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lower-+.f64 89.6

         \[\leadsto \left(3 - \frac{\left(0.125 \cdot \left(3 - \color{blue}{\left(v + v\right)}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   9. Applied rewrites 89.6%

      \[\leadsto \left(3 - \frac{\left(0.125 \cdot \left(3 - \color{blue}{\left(v + v\right)}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{\color{blue}{r \cdot r}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      12. lift-*.f64 99.7

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      2. pow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r}}{r} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-3}{2} \cdot r, r, 2\right)}{r}}{r} \]

      8. lower-*.f64 99.8

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \]

   9. Applied rewrites 99.8%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(0.125 \cdot 3\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -1.5)
   (- (- 3.0 (/ (* (* 0.125 3.0) (* (* r w) (* r w))) 1.0)) 4.5)
   (/ (/ (fma (* -1.5 r) r 2.0) r) r)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - (((0.125 * 3.0) * ((r * w) * (r * w))) / 1.0)) - 4.5;
	} else {
		tmp = (fma((-1.5 * r), r, 2.0) / r) / r;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.125 * 3.0) * Float64(Float64(r * w) * Float64(r * w))) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(fma(Float64(-1.5 * r), r, 2.0) / r) / r);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5], N[(N[(3.0 - N[(N[(N[(0.125 * 3.0), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 85.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 92.6

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 92.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.6%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{\color{blue}{1}}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.8%

       \[\leadsto \left(3 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{\color{blue}{1}}\right) - 4.5 \]

   11. Taylor expanded in v around 0

       \[\leadsto \left(3 - \frac{\left(\frac{1}{8} \cdot \color{blue}{3}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1}\right) - \frac{9}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 84.0%

       \[\leadsto \left(3 - \frac{\left(0.125 \cdot \color{blue}{3}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1}\right) - 4.5 \]

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 87.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{\color{blue}{r \cdot r}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      12. lift-*.f64 99.7

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      2. pow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r}}{r} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-3}{2} \cdot r, r, 2\right)}{r}}{r} \]

      8. lower-*.f64 99.8

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \]

   9. Applied rewrites 99.8%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{-111}:\\ \;\;\;\;\left(t\_0 - \frac{0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot r\right) \cdot \frac{w \cdot r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= r 2e-111)
     (- (- t_0 (/ (* 0.375 (* (* r w) (* r w))) (- 1.0 v))) 4.5)
     (-
      (- t_0 (* (* (* (* (fma v -2.0 3.0) 0.125) w) r) (/ (* w r) (- 1.0 v))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (r <= 2e-111) {
		tmp = (t_0 - ((0.375 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((((fma(v, -2.0, 3.0) * 0.125) * w) * r) * ((w * r) / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 2e-111)
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * w) * r) * Float64(Float64(w * r) / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2e-111], N[(N[(t$95$0 - N[(N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.00000000000000018e-111

   1. Initial program 84.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 96.9

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 96.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   if 2.00000000000000018e-111 < r

   1. Initial program 89.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 93.6

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 93.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \color{blue}{\left(3 - 2 \cdot v\right)}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{2 \cdot v}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right)} \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right)} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right)} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(3 + \color{blue}{-2} \cdot v\right) \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      15. +-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(-2 \cdot v + 3\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\mathsf{fma}\left(-2, v, 3\right)} \cdot \frac{1}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      17. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(r \cdot w\right)}{1 - v}\right) - \frac{9}{2} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      20. lift-*.f64 93.6

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - 4.5 \]

   6. Applied rewrites 93.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - 4.5 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{\color{blue}{1 - v}}\right) - \frac{9}{2} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right)} \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}\right)} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\color{blue}{\left(-2 \cdot v + 3\right)} \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(-2 \cdot v + 3\right) \cdot \frac{1}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(-2 \cdot v + 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{w \cdot r}{1 - v}}\right) - \frac{9}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(-2 \cdot v + 3\right) \cdot \frac{1}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{w \cdot r}{1 - v}}\right) - \frac{9}{2} \]

   8. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot r\right) \cdot \frac{w \cdot r}{1 - v}}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;r \leq 4:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot t\_0}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(0.125 \cdot \left(3 - \left(v + v\right)\right)\right) \cdot t\_0}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))))
   (if (<= r 4.0)
     (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* 0.375 t_0) (- 1.0 v))) 4.5)
     (- (- 3.0 (/ (* (* 0.125 (- 3.0 (+ v v))) t_0) (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= 4.0) {
		tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 * t_0) / (1.0 - v))) - 4.5;
	} else {
		tmp = (3.0 - (((0.125 * (3.0 - (v + v))) * t_0) / (1.0 - v))) - 4.5;
	}
	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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    if (r <= 4.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) - ((0.375d0 * t_0) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (3.0d0 - (((0.125d0 * (3.0d0 - (v + v))) * t_0) / (1.0d0 - v))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= 4.0) {
		tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 * t_0) / (1.0 - v))) - 4.5;
	} else {
		tmp = (3.0 - (((0.125 * (3.0 - (v + v))) * t_0) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	tmp = 0
	if r <= 4.0:
		tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 * t_0) / (1.0 - v))) - 4.5
	else:
		tmp = (3.0 - (((0.125 * (3.0 - (v + v))) * t_0) / (1.0 - v))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (r <= 4.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.375 * t_0) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v + v))) * t_0) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	tmp = 0.0;
	if (r <= 4.0)
		tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 * t_0) / (1.0 - v))) - 4.5;
	else
		tmp = (3.0 - (((0.125 * (3.0 - (v + v))) * t_0) / (1.0 - v))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4.0], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(0.125 * N[(3.0 - N[(v + v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4

   1. Initial program 84.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 95.8

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 95.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.4%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   if 4 < r

   1. Initial program 89.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{{w}^{2}} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      6. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{{r}^{2}}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}}{1 - v}\right) - \frac{9}{2} \]

      8. pow-prod-down N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{2}}}{1 - v}\right) - \frac{9}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-*.f64 95.5

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 95.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.1%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{2 \cdot v}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      2. count-2-rev N/A

         \[\leadsto \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - \color{blue}{\left(v + v\right)}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      3. lower-+.f64 95.1

         \[\leadsto \left(3 - \frac{\left(0.125 \cdot \left(3 - \color{blue}{\left(v + v\right)}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   9. Applied rewrites 95.1%

      \[\leadsto \left(3 - \frac{\left(0.125 \cdot \left(3 - \color{blue}{\left(v + v\right)}\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 100000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 100000000.0) (/ (fma (* -1.5 r) r 2.0) (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 100000000.0) {
		tmp = fma((-1.5 * r), r, 2.0) / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 100000000.0)
		tmp = Float64(fma(Float64(-1.5 * r), r, 2.0) / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 100000000.0], N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1e8

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 67.9

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2} \cdot r, r, 2\right)}{\color{blue}{r} \cdot r} \]

      5. lower-*.f64 67.8

         \[\leadsto \frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r} \]

   7. Applied rewrites 67.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{\color{blue}{r} \cdot r} \]

   if 1e8 < r

   1. Initial program 89.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 15.2

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 15.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.9%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 200000000000.0) (/ (fma -1.5 (* r r) 2.0) (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 200000000000.0) {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 200000000000.0)
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 200000000000.0], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 2e11

   1. Initial program 85.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 67.7

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   if 2e11 < r

   1. Initial program 89.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 14.0

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 14.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.0%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ (/ 2.0 r) r) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = (2.0d0 / r) / r
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 84.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 68.0

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{\color{blue}{r \cdot r}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot r} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{\color{blue}{r} \cdot r} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{\color{blue}{r}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r}}{r} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{2} \cdot {r}^{2} + 2}{r}}{r} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{r}^{2} \cdot \frac{-3}{2} + 2}{r}}{r} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({r}^{2}, \frac{-3}{2}, 2\right)}{r}}{r} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, \frac{-3}{2}, 2\right)}{r}}{r} \]

      12. lift-*.f64 79.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \]

   7. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}} \]

   8. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{2}{r}}{r} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.9%

       \[\leadsto \frac{\frac{2}{r}}{r} \]

   if 1.1499999999999999 < r

   1. Initial program 89.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 16.3

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.3%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 84.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 68.0

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around 0

      \[\leadsto \frac{2}{\color{blue}{r} \cdot r} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.9%

      \[\leadsto \frac{2}{\color{blue}{r} \cdot r} \]

   if 1.1499999999999999 < r

   1. Initial program 89.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

      7. lift-*.f64 16.3

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   5. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

   6. Taylor expanded in r around inf

      \[\leadsto \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.3%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 13.7% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 86.2%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

   4. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

   6. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   7. lift-*.f64 54.7

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

5. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

6. Taylor expanded in r around inf

   \[\leadsto \frac{-3}{2} \]
7. Step-by-step derivation

8. Applied rewrites 14.8%

   \[\leadsto -1.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025056 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))