Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Percentage Accurate: 90.6% → 99.3%

Time: 3.4s

Alternatives: 9

Speedup: 1.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.6% accurate, 1.0× speedup

Math

\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, \left|x\right| \cdot \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(z, \frac{z}{\left|x\right|}, \frac{-t}{\left|x\right|}\right), \left|x\right|\right) \cdot \left|x\right|\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs x) 5e+120)
   (fma (fma t y (* (* z (- y)) z)) 4.0 (* (fabs x) (fabs x)))
   (*
    (fma (* -4.0 y) (fma z (/ z (fabs x)) (/ (- t) (fabs x))) (fabs x))
    (fabs x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(x) <= 5e+120) {
		tmp = fma(fma(t, y, ((z * -y) * z)), 4.0, (fabs(x) * fabs(x)));
	} else {
		tmp = fma((-4.0 * y), fma(z, (z / fabs(x)), (-t / fabs(x))), fabs(x)) * fabs(x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(x) <= 5e+120)
		tmp = fma(fma(t, y, Float64(Float64(z * Float64(-y)) * z)), 4.0, Float64(abs(x) * abs(x)));
	else
		tmp = Float64(fma(Float64(-4.0 * y), fma(z, Float64(z / abs(x)), Float64(Float64(-t) / abs(x))), abs(x)) * abs(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e+120], N[(N[(t * y + N[(N[(z * (-y)), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * N[(z * N[(z / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[((-t) / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000019e120

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) + x \cdot x} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      6. sub-flip N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

   3. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, x \cdot x\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + x \cdot x} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) - \left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      10. lower-fma.f64 93.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, 4, x \cdot x\right)} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, x \cdot x\right)} \]

   if 5.00000000000000019e120 < x

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      3. lift--.f64 N/A

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      4. sub-flip N/A

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)} \]

      6. associate--r+ N/A

         \[\leadsto \color{blue}{\left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) - \left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) - \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(t\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(y \cdot 4\right)\right)} + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      11. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)\right)} + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(y \cdot 4\right)}\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(4 \cdot y\right)}\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right) \cdot y}\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      15. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right)\right) \cdot y} + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right), y, x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right)} \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 4, y, \mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, x \cdot x\right)\right)} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\left(x - -4 \cdot \frac{\left(t - z \cdot z\right) \cdot y}{x}\right) \cdot x} \]

   7. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{x}, x\right)} \cdot x \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \color{blue}{\frac{z \cdot z - t}{x}}, x\right) \cdot x \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \color{blue}{\frac{\mathsf{neg}\left(\left(z \cdot z - t\right)\right)}{\mathsf{neg}\left(x\right)}}, x\right) \cdot x \]

      3. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \frac{\mathsf{neg}\left(\color{blue}{\left(z \cdot z - t\right)}\right)}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      4. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \frac{\color{blue}{t - z \cdot z}}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      5. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \frac{\color{blue}{t + \left(\mathsf{neg}\left(z \cdot z\right)\right)}}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot z\right)\right) + t}}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      7. div-add N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \color{blue}{\frac{\mathsf{neg}\left(z \cdot z\right)}{\mathsf{neg}\left(x\right)} + \frac{t}{\mathsf{neg}\left(x\right)}}, x\right) \cdot x \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \color{blue}{\frac{z \cdot z}{x}} + \frac{t}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot y, \frac{\color{blue}{z \cdot z}}{x} + \frac{t}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \color{blue}{z \cdot \frac{z}{x}} + \frac{t}{\mathsf{neg}\left(x\right)}, x\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \color{blue}{\mathsf{fma}\left(z, \frac{z}{x}, \frac{t}{\mathsf{neg}\left(x\right)}\right)}, x\right) \cdot x \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(z, \color{blue}{\frac{z}{x}}, \frac{t}{\mathsf{neg}\left(x\right)}\right), x\right) \cdot x \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(z, \frac{z}{x}, \color{blue}{\mathsf{neg}\left(\frac{t}{x}\right)}\right), x\right) \cdot x \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(z, \frac{z}{x}, \color{blue}{\frac{\mathsf{neg}\left(t\right)}{x}}\right), x\right) \cdot x \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(z, \frac{z}{x}, \color{blue}{\frac{\mathsf{neg}\left(t\right)}{x}}\right), x\right) \cdot x \]

      16. lower-neg.f64 83.3

          \[\leadsto \mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(z, \frac{z}{x}, \frac{\color{blue}{-t}}{x}\right), x\right) \cdot x \]

   9. Applied rewrites 83.3%

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

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, \left|x\right| \cdot \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{\left|x\right|}, \left|x\right|\right) \cdot \left|x\right|\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs x) 5e+120)
   (fma (fma t y (* (* z (- y)) z)) 4.0 (* (fabs x) (fabs x)))
   (* (fma (* -4.0 y) (/ (- (* z z) t) (fabs x)) (fabs x)) (fabs x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(x) <= 5e+120) {
		tmp = fma(fma(t, y, ((z * -y) * z)), 4.0, (fabs(x) * fabs(x)));
	} else {
		tmp = fma((-4.0 * y), (((z * z) - t) / fabs(x)), fabs(x)) * fabs(x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(x) <= 5e+120)
		tmp = fma(fma(t, y, Float64(Float64(z * Float64(-y)) * z)), 4.0, Float64(abs(x) * abs(x)));
	else
		tmp = Float64(fma(Float64(-4.0 * y), Float64(Float64(Float64(z * z) - t) / abs(x)), abs(x)) * abs(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e+120], N[(N[(t * y + N[(N[(z * (-y)), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000019e120

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) + x \cdot x} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      6. sub-flip N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

   3. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, x \cdot x\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + x \cdot x} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) - \left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      10. lower-fma.f64 93.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, 4, x \cdot x\right)} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, x \cdot x\right)} \]

   if 5.00000000000000019e120 < x

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      3. lift--.f64 N/A

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      4. sub-flip N/A

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)} \]

      6. associate--r+ N/A

         \[\leadsto \color{blue}{\left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) - \left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) - \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(t\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(y \cdot 4\right)\right)} + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      11. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(y \cdot 4\right)\right)\right)} + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(y \cdot 4\right)}\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(4 \cdot y\right)}\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right) \cdot y}\right)\right) + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      15. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right)\right) \cdot y} + \left(x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right) \cdot 4\right), y, x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right)} \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 4, y, \mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, x \cdot x\right)\right)} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\left(x - -4 \cdot \frac{\left(t - z \cdot z\right) \cdot y}{x}\right) \cdot x} \]

   7. Applied rewrites 83.0%

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

Alternative 3: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5e+43)
   (fma (fma t y (* (* z (- y)) z)) 4.0 (* x x))
   (fma x x (* (* (- t (* z z)) y) 4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5e+43) {
		tmp = fma(fma(t, y, ((z * -y) * z)), 4.0, (x * x));
	} else {
		tmp = fma(x, x, (((t - (z * z)) * y) * 4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5e+43)
		tmp = fma(fma(t, y, Float64(Float64(z * Float64(-y)) * z)), 4.0, Float64(x * x));
	else
		tmp = fma(x, x, Float64(Float64(Float64(t - Float64(z * z)) * y) * 4.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5e+43], N[(N[(t * y + N[(N[(z * (-y)), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000004e43

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) + x \cdot x} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      6. sub-flip N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

   3. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, x \cdot x\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + x \cdot x} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) - \left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      10. lower-fma.f64 93.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, 4, x \cdot x\right)} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, x \cdot x\right)} \]

   if 5.0000000000000004e43 < y

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)}\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\left(z \cdot z - t\right)\right)\right)}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \color{blue}{\left(t - z \cdot z\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right)} \cdot \left(t - z \cdot z\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot \left(t - z \cdot z\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      17. lower--.f64 93.0

          \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{\left(t - z \cdot z\right)} \cdot y\right) \cdot 4\right) \]

   3. Applied rewrites 93.0%

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

Alternative 4: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-255}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t - \left(z \cdot y\right) \cdot z, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t, \left(\left(-z\right) \cdot z\right) \cdot y\right) \cdot 4\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.6e-255)
   (fma (- (* y t) (* (* z y) z)) 4.0 (* x x))
   (fma x x (* (fma y t (* (* (- z) z) y)) 4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.6e-255) {
		tmp = fma(((y * t) - ((z * y) * z)), 4.0, (x * x));
	} else {
		tmp = fma(x, x, (fma(y, t, ((-z * z) * y)) * 4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.6e-255)
		tmp = fma(Float64(Float64(y * t) - Float64(Float64(z * y) * z)), 4.0, Float64(x * x));
	else
		tmp = fma(x, x, Float64(fma(y, t, Float64(Float64(Float64(-z) * z) * y)) * 4.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.6e-255], N[(N[(N[(y * t), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(y * t + N[(N[((-z) * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.60000000000000023e-255

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) + x \cdot x} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      6. sub-flip N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

   3. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, x \cdot x\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + x \cdot x} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) - \left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      10. lower-fma.f64 93.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, 4, x \cdot x\right)} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, x \cdot x\right)} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y + \left(z \cdot \left(-y\right)\right) \cdot z}, 4, x \cdot x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y} + \left(z \cdot \left(-y\right)\right) \cdot z, 4, x \cdot x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y + \color{blue}{\left(z \cdot \left(-y\right)\right) \cdot z}, 4, x \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z}, 4, x \cdot x\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z}, 4, x \cdot x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y} - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z, 4, x \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot t} - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z, 4, x \cdot x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot t} - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z, 4, x \cdot x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot t - \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z}, 4, x \cdot x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \left(\mathsf{neg}\left(\color{blue}{z \cdot \left(-y\right)}\right)\right) \cdot z, 4, x \cdot x\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \left(\mathsf{neg}\left(z \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot z, 4, x \cdot x\right) \]

      12. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot y\right)\right)}\right)\right) \cdot z, 4, x \cdot x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \color{blue}{\left(z \cdot y\right)} \cdot z, 4, x \cdot x\right) \]

      14. lower-*.f64 93.7

          \[\leadsto \mathsf{fma}\left(y \cdot t - \color{blue}{\left(z \cdot y\right)} \cdot z, 4, x \cdot x\right) \]

   7. Applied rewrites 93.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot t - \left(z \cdot y\right) \cdot z}, 4, x \cdot x\right) \]

   if -5.60000000000000023e-255 < t

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)}\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\left(z \cdot z - t\right)\right)\right)}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \color{blue}{\left(t - z \cdot z\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right)} \cdot \left(t - z \cdot z\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot \left(t - z \cdot z\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      17. lower--.f64 93.0

          \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{\left(t - z \cdot z\right)} \cdot y\right) \cdot 4\right) \]

   3. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right)} \cdot 4\right) \]

      3. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot \color{blue}{\left(t - z \cdot z\right)}\right) \cdot 4\right) \]

      4. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(z \cdot z\right)\right)\right)}\right) \cdot 4\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(t \cdot y + \left(\mathsf{neg}\left(z \cdot z\right)\right) \cdot y\right)} \cdot 4\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{y \cdot t} + \left(\mathsf{neg}\left(z \cdot z\right)\right) \cdot y\right) \cdot 4\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(y, t, \left(\mathsf{neg}\left(z \cdot z\right)\right) \cdot y\right)} \cdot 4\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t, \color{blue}{\left(\mathsf{neg}\left(z \cdot z\right)\right) \cdot y}\right) \cdot 4\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t, \left(\mathsf{neg}\left(\color{blue}{z \cdot z}\right)\right) \cdot y\right) \cdot 4\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot z\right)} \cdot y\right) \cdot 4\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot z\right)} \cdot y\right) \cdot 4\right) \]

      12. lower-neg.f64 92.7

          \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, t, \left(\color{blue}{\left(-z\right)} \cdot z\right) \cdot y\right) \cdot 4\right) \]

   5. Applied rewrites 92.7%

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

Alternative 5: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t - \left(z \cdot y\right) \cdot z, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5e-230)
   (fma (- (* y t) (* (* z y) z)) 4.0 (* x x))
   (fma x x (* (* (- t (* z z)) y) 4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5e-230) {
		tmp = fma(((y * t) - ((z * y) * z)), 4.0, (x * x));
	} else {
		tmp = fma(x, x, (((t - (z * z)) * y) * 4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5e-230)
		tmp = fma(Float64(Float64(y * t) - Float64(Float64(z * y) * z)), 4.0, Float64(x * x));
	else
		tmp = fma(x, x, Float64(Float64(Float64(t - Float64(z * z)) * y) * 4.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5e-230], N[(N[(N[(y * t), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000000035e-230

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) + x \cdot x} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      6. sub-flip N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

   3. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, x \cdot x\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + x \cdot x} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) - \left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{4 \cdot \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y\right) \cdot 4 + \color{blue}{x \cdot x} \]

      10. lower-fma.f64 93.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-y\right) \cdot z\right) \cdot z - \left(-t\right) \cdot y, 4, x \cdot x\right)} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, y, \left(z \cdot \left(-y\right)\right) \cdot z\right), 4, x \cdot x\right)} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y + \left(z \cdot \left(-y\right)\right) \cdot z}, 4, x \cdot x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y} + \left(z \cdot \left(-y\right)\right) \cdot z, 4, x \cdot x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y + \color{blue}{\left(z \cdot \left(-y\right)\right) \cdot z}, 4, x \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z}, 4, x \cdot x\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z}, 4, x \cdot x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y} - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z, 4, x \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot t} - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z, 4, x \cdot x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot t} - \left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z, 4, x \cdot x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot t - \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-y\right)\right)\right) \cdot z}, 4, x \cdot x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \left(\mathsf{neg}\left(\color{blue}{z \cdot \left(-y\right)}\right)\right) \cdot z, 4, x \cdot x\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \left(\mathsf{neg}\left(z \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot z, 4, x \cdot x\right) \]

      12. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot y\right)\right)}\right)\right) \cdot z, 4, x \cdot x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y \cdot t - \color{blue}{\left(z \cdot y\right)} \cdot z, 4, x \cdot x\right) \]

      14. lower-*.f64 93.7

          \[\leadsto \mathsf{fma}\left(y \cdot t - \color{blue}{\left(z \cdot y\right)} \cdot z, 4, x \cdot x\right) \]

   7. Applied rewrites 93.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot t - \left(z \cdot y\right) \cdot z}, 4, x \cdot x\right) \]

   if -5.00000000000000035e-230 < t

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)}\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\left(z \cdot z - t\right)\right)\right)}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \color{blue}{\left(t - z \cdot z\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right)} \cdot \left(t - z \cdot z\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot \left(t - z \cdot z\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

      17. lower--.f64 93.0

          \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{\left(t - z \cdot z\right)} \cdot y\right) \cdot 4\right) \]

   3. Applied rewrites 93.0%

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

Alternative 6: 93.0% accurate, 1.0× speedup

Math

\[\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right) \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x x (* (* (- t (* z z)) y) 4.0)))

C

double code(double x, double y, double z, double t) {
	return fma(x, x, (((t - (z * z)) * y) * 4.0));
}

Julia

function code(x, y, z, t)
	return fma(x, x, Float64(Float64(Float64(t - Float64(z * z)) * y) * 4.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(x * x + N[(N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

      \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)\right)} \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)}\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\left(z \cdot z - t\right)\right)\right)}\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

   9. sub-negate-rev N/A

      \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \color{blue}{\left(t - z \cdot z\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right)} \cdot \left(t - z \cdot z\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot \left(t - z \cdot z\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

   17. lower--.f64 93.0

       \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{\left(t - z \cdot z\right)} \cdot y\right) \cdot 4\right) \]

3. Applied rewrites 93.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)} \]
4. Add Preprocessing

Alternative 7: 67.5% accurate, 1.5× speedup

Math

\[\mathsf{fma}\left(x, x, \left(t \cdot y\right) \cdot 4\right) \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x x (* (* t y) 4.0)))

C

double code(double x, double y, double z, double t) {
	return fma(x, x, ((t * y) * 4.0));
}

Julia

function code(x, y, z, t)
	return fma(x, x, Float64(Float64(t * y) * 4.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(x * x + N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

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

      \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)\right)} \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)}\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\left(z \cdot z - t\right)\right)\right)}\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

   9. sub-negate-rev N/A

      \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \color{blue}{\left(t - z \cdot z\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right)} \cdot \left(t - z \cdot z\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot \left(t - z \cdot z\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot \left(t - z \cdot z\right)\right) \cdot 4}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(t - z \cdot z\right) \cdot y\right)} \cdot 4\right) \]

   17. lower--.f64 93.0

       \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{\left(t - z \cdot z\right)} \cdot y\right) \cdot 4\right) \]

3. Applied rewrites 93.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)} \]

4. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{t} \cdot y\right) \cdot 4\right) \]
5. Step-by-step derivation

6. Applied rewrites 67.5%

   \[\leadsto \mathsf{fma}\left(x, x, \left(\color{blue}{t} \cdot y\right) \cdot 4\right) \]
7. Add Preprocessing

Alternative 8: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 32000000000:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left|x\right| \cdot \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs x) 32000000000.0)
   (* 4.0 (* t y))
   (* 1.0 (* (fabs x) (fabs x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(x) <= 32000000000.0) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = 1.0 * (fabs(x) * fabs(x));
	}
	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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (abs(x) <= 32000000000.0d0) then
        tmp = 4.0d0 * (t * y)
    else
        tmp = 1.0d0 * (abs(x) * abs(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.abs(x) <= 32000000000.0) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = 1.0 * (Math.abs(x) * Math.abs(x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.fabs(x) <= 32000000000.0:
		tmp = 4.0 * (t * y)
	else:
		tmp = 1.0 * (math.fabs(x) * math.fabs(x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(x) <= 32000000000.0)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(1.0 * Float64(abs(x) * abs(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (abs(x) <= 32000000000.0)
		tmp = 4.0 * (t * y);
	else
		tmp = 1.0 * (abs(x) * abs(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[x], $MachinePrecision], 32000000000.0], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e10

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 4 \cdot \color{blue}{\left(t \cdot y\right)} \]

      2. lower-*.f64 31.6

         \[\leadsto 4 \cdot \left(t \cdot \color{blue}{y}\right) \]

   4. Applied rewrites 31.6%

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]

   if 3.2e10 < x

   1. Initial program 90.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)}{x \cdot x}\right) \cdot \left(x \cdot x\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)}{x \cdot x}\right) \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\left(1 - \frac{\left(\left(t - z \cdot z\right) \cdot y\right) \cdot -4}{x \cdot x}\right) \cdot \left(x \cdot x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(x \cdot x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 41.5%

      \[\leadsto \color{blue}{1} \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 31.6% accurate, 2.6× speedup

Math

\[4 \cdot \left(t \cdot y\right) \]

FPCore

(FPCore (x y z t) :precision binary64 (* 4.0 (* t y)))

C

double code(double x, double y, double z, double t) {
	return 4.0 * (t * y);
}

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(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 4.0d0 * (t * y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 4.0 * (t * y);
}

Python

def code(x, y, z, t):
	return 4.0 * (t * y)

Julia

function code(x, y, z, t)
	return Float64(4.0 * Float64(t * y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 4.0 * (t * y);
end

Wolfram

code[x_, y_, z_, t_] := N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 4 \cdot \color{blue}{\left(t \cdot y\right)} \]

   2. lower-*.f64 31.6

      \[\leadsto 4 \cdot \left(t \cdot \color{blue}{y}\right) \]

4. Applied rewrites 31.6%

   \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64
  (- (* x x) (* (* y 4.0) (- (* z z) t))))