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

Percentage Accurate: 91.0% → 95.5%

Time: 3.0s

Alternatives: 6

Speedup: 0.7×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

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: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

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: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.2e+246)
   (fma (* (* -4.0 y) z) z (fma (* t y) 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 <= 6.2e+246) {
		tmp = fma(((-4.0 * y) * z), z, fma((t * y), 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 <= 6.2e+246)
		tmp = fma(Float64(Float64(-4.0 * y) * z), z, fma(Float64(t * y), 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, 6.2e+246], N[(N[(N[(-4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $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 < 6.19999999999999977e246

   1. Initial program 91.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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-rgt-in N/A

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

      8. associate-+l+ N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*l* N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 94.7%

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

   if 6.19999999999999977e246 < y

   1. Initial program 91.0%

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

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

      4. associate-*l* N/A

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

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

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

      6. associate-*l* N/A

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

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

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

      8. lift-*.f64 N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.2%

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

Alternative 2: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, 4 \cdot \left(t \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.4e-96)
   (fma (* (* -4.0 y) z) z (* 4.0 (* t y)))
   (if (<= x 1e+212)
     (fma x x (* (* (- t (* z z)) y) 4.0))
     (fma (* t 4.0) y (* x x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.4e-96) {
		tmp = fma(((-4.0 * y) * z), z, (4.0 * (t * y)));
	} else if (x <= 1e+212) {
		tmp = fma(x, x, (((t - (z * z)) * y) * 4.0));
	} else {
		tmp = fma((t * 4.0), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.4e-96)
		tmp = fma(Float64(Float64(-4.0 * y) * z), z, Float64(4.0 * Float64(t * y)));
	elseif (x <= 1e+212)
		tmp = fma(x, x, Float64(Float64(Float64(t - Float64(z * z)) * y) * 4.0));
	else
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 3.4e-96], N[(N[(N[(-4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+212], N[(x * x + N[(N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4000000000000001e-96

   1. Initial program 91.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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-rgt-in N/A

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

      8. associate-+l+ N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*l* N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 63.2

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

   6. Applied rewrites 63.2%

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

   if 3.4000000000000001e-96 < x < 9.9999999999999991e211

   1. Initial program 91.0%

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

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

      4. associate-*l* N/A

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

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

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

      6. associate-*l* N/A

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

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

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

      8. lift-*.f64 N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.2%

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

   if 9.9999999999999991e211 < x

   1. Initial program 91.0%

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

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

      4. associate-*l* N/A

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

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

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

      6. associate-*l* N/A

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

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

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

      8. lift-*.f64 N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.2%

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

   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 67.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

      6. sqr-neg-rev N/A

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

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

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

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

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

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

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

   8. Applied rewrites 68.1%

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

Alternative 3: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1e+212)
   (fma x x (* (* (- t (* z z)) y) 4.0))
   (fma (* t 4.0) y (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1e+212) {
		tmp = fma(x, x, (((t - (z * z)) * y) * 4.0));
	} else {
		tmp = fma((t * 4.0), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1e+212)
		tmp = fma(x, x, Float64(Float64(Float64(t - Float64(z * z)) * y) * 4.0));
	else
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1e+212], N[(x * x + N[(N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999991e211

   1. Initial program 91.0%

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

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

      4. associate-*l* N/A

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

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

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

      6. associate-*l* N/A

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

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

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

      8. lift-*.f64 N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.2%

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

   if 9.9999999999999991e211 < x

   1. Initial program 91.0%

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

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

      4. associate-*l* N/A

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

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

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

      6. associate-*l* N/A

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

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

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

      8. lift-*.f64 N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.2%

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

   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 67.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

      6. sqr-neg-rev N/A

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

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

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

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

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

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

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

   8. Applied rewrites 68.1%

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

Alternative 4: 68.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

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

   4. associate-*l* N/A

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

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

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

   6. associate-*l* N/A

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

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

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

   8. lift-*.f64 N/A

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

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

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

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

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. associate-*l* N/A

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

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

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

3. Applied rewrites 93.2%

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

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 67.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-abs-rev N/A

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

   6. sqr-neg-rev N/A

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

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

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

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

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

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

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

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

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

   16. sqr-neg-rev N/A

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

   17. sqr-abs-rev N/A

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

   18. lift-*.f64 N/A

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

   19. remove-double-neg N/A

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

8. Applied rewrites 68.1%

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

Alternative 5: 67.6% accurate, 1.5× speedup

Math

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

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 91.0%

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

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

   4. associate-*l* N/A

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

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

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

   6. associate-*l* N/A

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

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

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

   8. lift-*.f64 N/A

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

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

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

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

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. associate-*l* N/A

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

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

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

3. Applied rewrites 93.2%

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

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 67.6%

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

Alternative 6: 31.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(t \cdot y\right) \end{array} \]

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 91.0%

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

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 31.7

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

4. Applied rewrites 31.7%

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

Reproduce

herbie shell --seed 2025149 
(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))))