Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.7%

Time: 5.8s

Alternatives: 21

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: 84.8% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, 4.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ (/ 2.0 (* r r)) 3.0)
  (fma (fma -0.25 v 0.375) (/ (/ 1.0 (pow (* w r) -2.0)) (- 1.0 v)) 4.5)))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - fma(fma(-0.25, v, 0.375), ((1.0 / pow((w * r), -2.0)) / (1.0 - v)), 4.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - fma(fma(-0.25, v, 0.375), Float64(Float64(1.0 / (Float64(w * r) ^ -2.0)) / Float64(1.0 - v)), 4.5))
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(1.0 / N[Power[N[(w * r), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

3. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in v around 0

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\frac{3}{8} + \frac{-1}{4} \cdot v}, \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, \frac{9}{2}\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\frac{-1}{4} \cdot v + \color{blue}{\frac{3}{8}}, \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, \frac{9}{2}\right) \]

   2. lower-fma.f64 99.7

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right), \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, 4.5\right) \]

8. Applied rewrites 99.7%

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)}, \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, 4.5\right) \]
9. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{0.375}{v} - 0.25\right) \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ (/ 2.0 (* r r)) 3.0)
  (fma (* (- (/ 0.375 v) 0.25) v) (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5)))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - fma((((0.375 / v) - 0.25) * v), (((r * w) * (r * w)) / (1.0 - v)), 4.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - fma(Float64(Float64(Float64(0.375 / v) - 0.25) * v), Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5))
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(0.375 / v), $MachinePrecision] - 0.25), $MachinePrecision] * v), $MachinePrecision] * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in v around inf

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{v \cdot \left(\frac{3}{8} \cdot \frac{1}{v} - \frac{1}{4}\right)}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{3}{8} \cdot \frac{1}{v} - \frac{1}{4}\right) \cdot \color{blue}{v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{3}{8} \cdot \frac{1}{v} - \frac{1}{4}\right) \cdot \color{blue}{v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]

   3. lower--.f64 N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{3}{8} \cdot \frac{1}{v} - \frac{1}{4}\right) \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{\frac{3}{8} \cdot 1}{v} - \frac{1}{4}\right) \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{\frac{3}{8}}{v} - \frac{1}{4}\right) \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, \frac{9}{2}\right) \]

   6. lower-/.f64 99.7

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\left(\frac{0.375}{v} - 0.25\right) \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\left(\frac{0.375}{v} - 0.25\right) \cdot v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]
7. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Applied rewrites 99.7%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}\\ t_1 := \left(\frac{\frac{2}{r}}{r} + 3\right) - \mathsf{fma}\left(-0.25 \cdot v, t\_0, 4.5\right)\\ \mathbf{if}\;v \leq -200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(0.375, t\_0, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (* (* r w) (* r w)) (- 1.0 v)))
        (t_1 (- (+ (/ (/ 2.0 r) r) 3.0) (fma (* -0.25 v) t_0 4.5))))
   (if (<= v -200000000.0)
     t_1
     (if (<= v 1.5) (- (+ (/ 2.0 (* r r)) 3.0) (fma 0.375 t_0 4.5)) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) / (1.0 - v);
	double t_1 = (((2.0 / r) / r) + 3.0) - fma((-0.25 * v), t_0, 4.5);
	double tmp;
	if (v <= -200000000.0) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((2.0 / (r * r)) + 3.0) - fma(0.375, t_0, 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))
	t_1 = Float64(Float64(Float64(Float64(2.0 / r) / r) + 3.0) - fma(Float64(-0.25 * v), t_0, 4.5))
	tmp = 0.0
	if (v <= -200000000.0)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - fma(0.375, t_0, 4.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(-0.25 * v), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -200000000.0], t$95$1, If[LessEqual[v, 1.5], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(0.375 * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e8 or 1.5 < v

   1. Initial program 82.1%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 99.6

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(-0.25 \cdot \color{blue}{v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   6. Applied rewrites 99.6%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{-0.25 \cdot v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

         \[\leadsto \left(\frac{\color{blue}{\frac{2}{r}}}{r} + 3\right) - \mathsf{fma}\left(-0.25 \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} + 3\right) - \mathsf{fma}\left(-0.25 \cdot v, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   if -2e8 < v < 1.5

   1. Initial program 87.3%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 99.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{0.375}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}\\ t_1 := \frac{2}{r \cdot r} + 3\\ t_2 := t\_1 - \mathsf{fma}\left(-0.25 \cdot v, t\_0, 4.5\right)\\ \mathbf{if}\;v \leq -200000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(0.375, t\_0, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (* (* r w) (* r w)) (- 1.0 v)))
        (t_1 (+ (/ 2.0 (* r r)) 3.0))
        (t_2 (- t_1 (fma (* -0.25 v) t_0 4.5))))
   (if (<= v -200000000.0)
     t_2
     (if (<= v 1.5) (- t_1 (fma 0.375 t_0 4.5)) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) / (1.0 - v);
	double t_1 = (2.0 / (r * r)) + 3.0;
	double t_2 = t_1 - fma((-0.25 * v), t_0, 4.5);
	double tmp;
	if (v <= -200000000.0) {
		tmp = t_2;
	} else if (v <= 1.5) {
		tmp = t_1 - fma(0.375, t_0, 4.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	t_2 = Float64(t_1 - fma(Float64(-0.25 * v), t_0, 4.5))
	tmp = 0.0
	if (v <= -200000000.0)
		tmp = t_2;
	elseif (v <= 1.5)
		tmp = Float64(t_1 - fma(0.375, t_0, 4.5));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(-0.25 * v), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -200000000.0], t$95$2, If[LessEqual[v, 1.5], N[(t$95$1 - N[(0.375 * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e8 or 1.5 < v

   1. Initial program 82.1%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 99.6

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(-0.25 \cdot \color{blue}{v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   6. Applied rewrites 99.6%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{-0.25 \cdot v}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]

   if -2e8 < v < 1.5

   1. Initial program 87.3%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 99.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{0.375}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot w\right) \cdot r\\ t_2 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_1 \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_2 \leq -1.5518272830176518:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot t\_1\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + 3\right) - \mathsf{fma}\left(0.375, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* (* w w) r))
        (t_2
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_1 r)) (- 1.0 v)))
          4.5)))
   (if (<= t_2 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_2 -1.5518272830176518)
       (- (- 3.0 (/ (* (* (* (fma v -2.0 3.0) 0.125) t_1) r) (- 1.0 v))) 4.5)
       (- (+ t_0 3.0) (fma 0.375 (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (t_1 * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_2 <= -1.5518272830176518) {
		tmp = (3.0 - ((((fma(v, -2.0, 3.0) * 0.125) * t_1) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 + 3.0) - fma(0.375, (((r * w) * (r * w)) / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(w * w) * r)
	t_2 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_1 * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_2 <= -1.5518272830176518)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * t_1) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 + 3.0) - fma(0.375, Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1.5518272830176518], N[(N[(3.0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$1), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -1.5518272830176518

   1. Initial program 98.4%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 97.6%

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

      1. +-commutative 97.6

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

   6. Applied rewrites 97.3%

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

   if -1.5518272830176518 < (-.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 84.0%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 97.2%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{0.375}, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 950:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(0.375, \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 950.0)
   (-
    (+ (/ 2.0 (* r r)) 3.0)
    (fma 0.375 (/ (* (* r w) (* r w)) (- 1.0 v)) 4.5))
   (-
    (- 3.0 (/ (* (* 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) {
	double tmp;
	if (r <= 950.0) {
		tmp = ((2.0 / (r * r)) + 3.0) - fma(0.375, (((r * w) * (r * w)) / (1.0 - v)), 4.5);
	} else {
		tmp = (3.0 - (((0.125 * (3.0 - (2.0 * v))) * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 950.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - fma(0.375, Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v)), 4.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(w * Float64(w * r)) * r)) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 950.0], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 950

   1. Initial program 83.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 88.9%

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

   if 950 < 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. Taylor expanded in r around inf

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

   4. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.5

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

   6. Applied rewrites 94.5%

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

Alternative 8: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot w\right) \cdot r\\ t_2 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_1 \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_2 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot t\_1\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* (* w w) r))
        (t_2
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_1 r)) (- 1.0 v)))
          4.5)))
   (if (<= t_2 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_2 -1.5)
       (- (- 3.0 (/ (* (* (* (fma v -2.0 3.0) 0.125) t_1) r) (- 1.0 v))) 4.5)
       (/ (fma -1.5 (* r r) 2.0) (* r r))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (t_1 * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_2 <= -1.5) {
		tmp = (3.0 - ((((fma(v, -2.0, 3.0) * 0.125) * t_1) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(w * w) * r)
	t_2 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_1 * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_2 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * t_1) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1.5], N[(N[(3.0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$1), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -1.5

   1. Initial program 87.0%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.7%

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

      1. +-commutative 86.7

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

   6. Applied rewrites 86.6%

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

   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 85.0%

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

   2. Taylor expanded in r around 0

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

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

   4. Applied rewrites 99.7%

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

Alternative 9: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_1 -1.5)
       (-
        (-
         3.0
         (/ (* (* (fma v -2.0 3.0) 0.125) (* (* (* w r) 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 t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - (((fma(v, -2.0, 3.0) * 0.125) * (((w * r) * 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)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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_1 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * Float64(Float64(Float64(w * r) * r) * w)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -1.5

   1. Initial program 87.0%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 86.7

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 88.7

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

   6. Applied rewrites 88.7%

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

   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 85.0%

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

   2. Taylor expanded in r around 0

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

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

   4. Applied rewrites 99.7%

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

Alternative 10: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ t_2 := \left(\left(3 + t\_0\right) - \frac{t\_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(3 - \frac{t\_1 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* 0.125 (- 3.0 (* 2.0 v))))
        (t_2
         (- (- (+ 3.0 t_0) (/ (* t_1 (* (* (* w w) r) r)) (- 1.0 v))) 4.5)))
   (if (<= t_2 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_2 -2e+44)
       (- (- 3.0 (/ (* t_1 (* (* w (* w r)) r)) 1.0)) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = ((3.0 + t_0) - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_2 <= -2e+44) {
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / 1.0)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.125 * (3.0 - (2.0 * v));
	double t_2 = ((3.0 + t_0) - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_2 <= -2e+44) {
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / 1.0)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 0.125 * (3.0 - (2.0 * v))
	t_2 = ((3.0 + t_0) - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_0 - ((((r * r) * 0.25) * w) * w)
	elif t_2 <= -2e+44:
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / 1.0)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	t_2 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(t_1 * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_2 <= -2e+44)
		tmp = Float64(Float64(3.0 - Float64(Float64(t_1 * Float64(Float64(w * Float64(w * r)) * r)) / 1.0)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 0.125 * (3.0 - (2.0 * v));
	t_2 = ((3.0 + t_0) - ((t_1 * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	elseif (t_2 <= -2e+44)
		tmp = (3.0 - ((t_1 * ((w * (w * r)) * r)) / 1.0)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - 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]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+44], N[(N[(3.0 - N[(N[(t$95$1 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -2.0000000000000002e44

   1. Initial program 98.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

   if -2.0000000000000002e44 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 93.0

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot w\right) \cdot r\\ t_2 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_1 \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot t\_1\right) \cdot r}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* (* w w) r))
        (t_2
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_1 r)) (- 1.0 v)))
          4.5)))
   (if (<= t_2 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_2 -2e+44)
       (- (- 3.0 (/ (* (* (* (fma v -2.0 3.0) 0.125) t_1) r) 1.0)) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (t_1 * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_2 <= -2e+44) {
		tmp = (3.0 - ((((fma(v, -2.0, 3.0) * 0.125) * t_1) * r) / 1.0)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(w * w) * r)
	t_2 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_1 * r)) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_2 <= -2e+44)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * t_1) * r) / 1.0)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+44], N[(N[(3.0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$1), $MachinePrecision] * r), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -2.0000000000000002e44

   1. Initial program 98.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 69.4%

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

   9. Applied rewrites 69.4%

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

   if -2.0000000000000002e44 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 93.0

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\left(\left(r \cdot r\right) \cdot \frac{\left(w \cdot w\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_1 -1e+50)
       (* (* (* r r) (/ (* (* w w) (fma v -2.0 3.0)) (- 1.0 v))) -0.125)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_1 <= -1e+50) {
		tmp = ((r * r) * (((w * w) * fma(v, -2.0, 3.0)) / (1.0 - v))) * -0.125;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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_1 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_1 <= -1e+50)
		tmp = Float64(Float64(Float64(r * r) * Float64(Float64(Float64(w * w) * fma(v, -2.0, 3.0)) / Float64(1.0 - v))) * -0.125);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+50], N[(N[(N[(r * r), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -1.0000000000000001e50

   1. Initial program 98.6%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in v around 0

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\frac{3}{8} + \frac{-1}{4} \cdot v}, \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, \frac{9}{2}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\frac{-1}{4} \cdot v + \color{blue}{\frac{3}{8}}, \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, \frac{9}{2}\right) \]

      2. lower-fma.f64 99.1

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, \color{blue}{v}, 0.375\right), \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, 4.5\right) \]

   8. Applied rewrites 99.1%

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)}, \frac{\frac{1}{{\left(w \cdot r\right)}^{-2}}}{1 - v}, 4.5\right) \]

   9. Taylor expanded in w around inf

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

   11. Applied rewrites 68.0%

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

   if -1.0000000000000001e50 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 92.9

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(\left(w \cdot w\right) \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot \left(r \cdot r\right)}{1 - v} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_1 -1e+50)
       (* (/ (* (* (* w w) (fma -2.0 v 3.0)) (* r r)) (- 1.0 v)) -0.125)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_1 <= -1e+50) {
		tmp = ((((w * w) * fma(-2.0, v, 3.0)) * (r * r)) / (1.0 - v)) * -0.125;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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_1 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_1 <= -1e+50)
		tmp = Float64(Float64(Float64(Float64(Float64(w * w) * fma(-2.0, v, 3.0)) * Float64(r * r)) / Float64(1.0 - v)) * -0.125);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+50], N[(N[(N[(N[(N[(w * w), $MachinePrecision] * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * N[(r * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -1.0000000000000001e50

   1. Initial program 98.6%

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

   2. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

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

   if -1.0000000000000001e50 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 92.9

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (* (* (* (* r r) 0.25) w) w))
     (if (<= t_1 -1e+50)
       (- (- 3.0 (* (* 0.375 (* r r)) (* w w))) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_1 <= -1e+50) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	} else if (t_1 <= -1e+50) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0 - ((((r * r) * 0.25) * w) * w)
	elif t_1 <= -1e+50:
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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_1 <= Float64(-Inf))
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w));
	elseif (t_1 <= -1e+50)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0 - ((((r * r) * 0.25) * w) * w);
	elseif (t_1 <= -1e+50)
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+50], N[(N[(3.0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w \]

      10. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left({r}^{2} \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      14. pow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \left(\left(\left(r \cdot r\right) \cdot \frac{1}{4}\right) \cdot w\right) \cdot w \]

      15. lift-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

   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)) < -1.0000000000000001e50

   1. Initial program 98.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.5

         \[\leadsto \left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot \color{blue}{w}\right)\right) - 4.5 \]

   7. Applied rewrites 51.5%

      \[\leadsto \left(3 - \color{blue}{\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)}\right) - 4.5 \]

   if -1.0000000000000001e50 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 92.9

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(3 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- (- 3.0 (* (* (* (* r r) 0.25) w) w)) 4.5)
     (if (<= t_1 -1e+50)
       (- (- 3.0 (* (* 0.375 (* r r)) (* w w))) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (3.0 - ((((r * r) * 0.25) * w) * w)) - 4.5;
	} else if (t_1 <= -1e+50) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - ((((r * r) * 0.25) * w) * w)) - 4.5;
	} else if (t_1 <= -1e+50) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (3.0 - ((((r * r) * 0.25) * w) * w)) - 4.5
	elif t_1 <= -1e+50:
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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_1 <= Float64(-Inf))
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(r * r) * 0.25) * w) * w)) - 4.5);
	elseif (t_1 <= -1e+50)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (3.0 - ((((r * r) * 0.25) * w) * w)) - 4.5;
	elseif (t_1 <= -1e+50)
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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$1, (-Infinity)], N[(N[(3.0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1e+50], N[(N[(3.0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 89.5

          \[\leadsto \left(3 - \left(\left(\left(r \cdot r\right) \cdot 0.25\right) \cdot w\right) \cdot w\right) - 4.5 \]

   10. Applied rewrites 89.5%

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

   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)) < -1.0000000000000001e50

   1. Initial program 98.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.5

         \[\leadsto \left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot \color{blue}{w}\right)\right) - 4.5 \]

   7. Applied rewrites 51.5%

      \[\leadsto \left(3 - \color{blue}{\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)}\right) - 4.5 \]

   if -1.0000000000000001e50 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 92.9

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (* (* -0.25 (* r r)) (* w w))
     (if (<= t_1 -1e+50)
       (- (- 3.0 (* (* 0.375 (* r r)) (* w w))) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else if (t_1 <= -1e+50) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else if (t_1 <= -1e+50) {
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (-0.25 * (r * r)) * (w * w)
	elif t_1 <= -1e+50:
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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_1 <= Float64(-Inf))
		tmp = Float64(Float64(-0.25 * Float64(r * r)) * Float64(w * w));
	elseif (t_1 <= -1e+50)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.375 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (-0.25 * (r * r)) * (w * w);
	elseif (t_1 <= -1e+50)
		tmp = (3.0 - ((0.375 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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$1, (-Infinity)], N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+50], N[(N[(3.0 - N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -inf.0

   1. Initial program 82.9%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      6. pow2 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      7. lift-*.f64 87.3

         \[\leadsto \left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

   9. Applied rewrites 87.3%

      \[\leadsto \left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \color{blue}{\left(w \cdot w\right)} \]

   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)) < -1.0000000000000001e50

   1. Initial program 98.6%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.5

         \[\leadsto \left(3 - \left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot \color{blue}{w}\right)\right) - 4.5 \]

   7. Applied rewrites 51.5%

      \[\leadsto \left(3 - \color{blue}{\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)}\right) - 4.5 \]

   if -1.0000000000000001e50 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 92.9

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5518272830176518:\\ \;\;\;\;\left(3 - \left(0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1.5518272830176518)
     (- (- 3.0 (* (* 0.25 (* r r)) (* w w))) 4.5)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5518272830176518) {
		tmp = (3.0 - ((0.25 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1.5518272830176518d0)) then
        tmp = (3.0d0 - ((0.25d0 * (r * r)) * (w * w))) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5518272830176518) {
		tmp = (3.0 - ((0.25 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5518272830176518:
		tmp = (3.0 - ((0.25 * (r * r)) * (w * w))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - 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) <= -1.5518272830176518)
		tmp = Float64(Float64(3.0 - Float64(Float64(0.25 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5518272830176518)
		tmp = (3.0 - ((0.25 * (r * r)) * (w * w))) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $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], -1.5518272830176518], N[(N[(3.0 - N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -1.5518272830176518

   1. Initial program 85.8%

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

   2. Taylor expanded in r around inf

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.2

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

   7. Applied rewrites 77.2%

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

   if -1.5518272830176518 < (-.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 84.0%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 94.6

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -1e+50)
     (* (* -0.25 (* r r)) (* w w))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+50) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1d+50)) then
        tmp = ((-0.25d0) * (r * r)) * (w * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+50) {
		tmp = (-0.25 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+50:
		tmp = (-0.25 * (r * r)) * (w * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - 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) <= -1e+50)
		tmp = Float64(Float64(-0.25 * Float64(r * r)) * Float64(w * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+50)
		tmp = (-0.25 * (r * r)) * (w * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $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], -1e+50], N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $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)) < -1.0000000000000001e50

   1. Initial program 85.5%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(0.25 \cdot \left(r \cdot r\right), w \cdot w, 1.5\right)} \]

   7. Taylor expanded in w around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot {w}^{2} \]

      6. pow2 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

      7. lift-*.f64 78.0

         \[\leadsto \left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right) \]

   9. Applied rewrites 78.0%

      \[\leadsto \left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot \color{blue}{\left(w \cdot w\right)} \]

   if -1.0000000000000001e50 < (-.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 84.3%

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

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 92.9

         \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} - 1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (- (/ 2.0 (* r r)) 1.5))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) - 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 = (2.0d0 / (r * r)) - 1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) - 1.5;
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) - 1.5

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) - 1.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) - 1.5;
end

Wolfram

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

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - \color{blue}{\frac{3}{2}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot 1}{{r}^{2}} - \frac{3}{2} \]

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 58.2

      \[\leadsto \frac{2}{r \cdot r} - 1.5 \]

4. Applied rewrites 58.2%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
5. Add Preprocessing

Alternative 20: 51.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.2:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.2) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.2:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.2)
		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 <= 1.2)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.2], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.19999999999999996

   1. Initial program 83.2%

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

   2. Taylor expanded in r around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 1.19999999999999996 < r

   1. Initial program 89.3%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in r around inf

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

   6. Applied rewrites 80.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{\left(w \cdot w\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, 0.125, \frac{1.5}{r \cdot r}\right) \cdot \left(r \cdot r\right)} \]

   7. Taylor expanded in w around 0

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

   9. Applied rewrites 27.2%

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

Alternative 21: 14.4% accurate, 41.6× 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 84.8%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in r around inf

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

6. Applied rewrites 43.9%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{\left(w \cdot w\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, 0.125, \frac{1.5}{r \cdot r}\right) \cdot \left(r \cdot r\right)} \]

7. Taylor expanded in w around 0

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

9. Applied rewrites 14.4%

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

Reproduce

herbie shell --seed 2025117 
(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))