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

Percentage Accurate: 91.5% → 95.8%

Time: 3.7s

Alternatives: 8

Speedup: 0.9×

• 49.7% 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: 91.5% 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: 95.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z\_m, z\_m, -t\right), -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(y - \frac{z\_m \cdot \left(z\_m \cdot y\right)}{t}\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 4.5e+153)
   (fma (fma z_m z_m (- t)) (* -4.0 y) (* x x))
   (- (* x x) (* (* -4.0 (- y (/ (* z_m (* z_m y)) t))) t))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.5e+153) {
		tmp = fma(fma(z_m, z_m, -t), (-4.0 * y), (x * x));
	} else {
		tmp = (x * x) - ((-4.0 * (y - ((z_m * (z_m * y)) / t))) * t);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 4.5e+153)
		tmp = fma(fma(z_m, z_m, Float64(-t)), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(-4.0 * Float64(y - Float64(Float64(z_m * Float64(z_m * y)) / t))) * t));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 4.5e+153], N[(N[(z$95$m * z$95$m + (-t)), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(-4.0 * N[(y - N[(N[(z$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.5000000000000001e153

   1. Initial program 93.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

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

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) + {x}^{2}} \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot \left({z}^{2} - t\right)} + {x}^{2} \]

      15. pow2 N/A

          \[\leadsto \left(-4 \cdot y\right) \cdot \left(\color{blue}{z \cdot z} - t\right) + {x}^{2} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)} + {x}^{2} \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z - t, -4 \cdot y, {x}^{2}\right)} \]

   4. Applied rewrites 95.6%

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

   if 4.5000000000000001e153 < z

   1. Initial program 77.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot x - \color{blue}{t \cdot \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

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

         \[\leadsto x \cdot x - \left(-4 \cdot y - \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      4. metadata-eval N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y - -4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      5. distribute-lft-out-- N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      11. pow2 N/A

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

      12. lift-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 2: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \left(4 \cdot t\right) \cdot y\\ \mathbf{if}\;z\_m \leq 2.6 \cdot 10^{-178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z\_m \leq 2.9 \cdot 10^{-111}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z\_m \leq 4 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m \cdot y\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (* (* 4.0 t) y)))
   (if (<= z_m 2.6e-178)
     t_1
     (if (<= z_m 2.9e-111)
       (* x x)
       (if (<= z_m 4e+21) t_1 (* (* (* z_m y) z_m) -4.0))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (4.0 * t) * y;
	double tmp;
	if (z_m <= 2.6e-178) {
		tmp = t_1;
	} else if (z_m <= 2.9e-111) {
		tmp = x * x;
	} else if (z_m <= 4e+21) {
		tmp = t_1;
	} else {
		tmp = ((z_m * y) * z_m) * -4.0;
	}
	return tmp;
}

Fortran

z_m =     private
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_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (4.0d0 * t) * y
    if (z_m <= 2.6d-178) then
        tmp = t_1
    else if (z_m <= 2.9d-111) then
        tmp = x * x
    else if (z_m <= 4d+21) then
        tmp = t_1
    else
        tmp = ((z_m * y) * z_m) * (-4.0d0)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (4.0 * t) * y;
	double tmp;
	if (z_m <= 2.6e-178) {
		tmp = t_1;
	} else if (z_m <= 2.9e-111) {
		tmp = x * x;
	} else if (z_m <= 4e+21) {
		tmp = t_1;
	} else {
		tmp = ((z_m * y) * z_m) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (4.0 * t) * y
	tmp = 0
	if z_m <= 2.6e-178:
		tmp = t_1
	elif z_m <= 2.9e-111:
		tmp = x * x
	elif z_m <= 4e+21:
		tmp = t_1
	else:
		tmp = ((z_m * y) * z_m) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(4.0 * t) * y)
	tmp = 0.0
	if (z_m <= 2.6e-178)
		tmp = t_1;
	elseif (z_m <= 2.9e-111)
		tmp = Float64(x * x);
	elseif (z_m <= 4e+21)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(z_m * y) * z_m) * -4.0);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (4.0 * t) * y;
	tmp = 0.0;
	if (z_m <= 2.6e-178)
		tmp = t_1;
	elseif (z_m <= 2.9e-111)
		tmp = x * x;
	elseif (z_m <= 4e+21)
		tmp = t_1;
	else
		tmp = ((z_m * y) * z_m) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z$95$m, 2.6e-178], t$95$1, If[LessEqual[z$95$m, 2.9e-111], N[(x * x), $MachinePrecision], If[LessEqual[z$95$m, 4e+21], t$95$1, N[(N[(N[(z$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.59999999999999998e-178 or 2.90000000000000002e-111 < z < 4e21

   1. Initial program 92.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto x \cdot \color{blue}{x} \]

      2. lift-*.f64 39.6

         \[\leadsto x \cdot \color{blue}{x} \]

   5. Applied rewrites 39.6%

      \[\leadsto \color{blue}{x \cdot x} \]

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 37.6

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

   8. Applied rewrites 37.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 37.6

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

   10. Applied rewrites 37.6%

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

   if 2.59999999999999998e-178 < z < 2.90000000000000002e-111

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto x \cdot \color{blue}{x} \]

      2. lift-*.f64 81.1

         \[\leadsto x \cdot \color{blue}{x} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{x \cdot x} \]

   if 4e21 < z

   1. Initial program 85.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot x - \color{blue}{t \cdot \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

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

         \[\leadsto x \cdot x - \left(-4 \cdot y - \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      4. metadata-eval N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y - -4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      5. distribute-lft-out-- N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      11. pow2 N/A

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

      12. lift-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 84.6

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

   10. Applied rewrites 84.6%

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

Alternative 3: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z\_m \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m \cdot z\_m\right) \cdot y, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m \cdot y\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 7.2e+21)
   (fma (* 4.0 t) y (* x x))
   (if (<= z_m 2e+151)
     (fma (* (* z_m z_m) y) -4.0 (* x x))
     (* (* (* z_m y) z_m) -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 7.2e+21) {
		tmp = fma((4.0 * t), y, (x * x));
	} else if (z_m <= 2e+151) {
		tmp = fma(((z_m * z_m) * y), -4.0, (x * x));
	} else {
		tmp = ((z_m * y) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 7.2e+21)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	elseif (z_m <= 2e+151)
		tmp = fma(Float64(Float64(z_m * z_m) * y), -4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z_m * y) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 7.2e+21], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2e+151], N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.2e21

   1. Initial program 93.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + 4 \cdot \left(\color{blue}{t} \cdot y\right) \]

      3. +-commutative N/A

         \[\leadsto 4 \cdot \left(t \cdot y\right) + \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot y\right) \cdot 4 + {\color{blue}{x}}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, \color{blue}{4}, {x}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

         \[\leadsto \left(4 \cdot t\right) \cdot y + {x}^{\color{blue}{2}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot t, \color{blue}{y}, {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot t, y, {x}^{2}\right) \]

      9. pow2 N/A

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

      10. lift-*.f64 74.8

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

   7. Applied rewrites 74.8%

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

   if 7.2e21 < z < 2.00000000000000003e151

   1. Initial program 95.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{{x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot {z}^{2}\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + -4 \cdot \left(\color{blue}{y} \cdot {z}^{2}\right) \]

      3. +-commutative N/A

         \[\leadsto -4 \cdot \left(y \cdot {z}^{2}\right) + \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot -4 + {\color{blue}{x}}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot {z}^{2}, \color{blue}{-4}, {x}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({z}^{2} \cdot y, -4, {x}^{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({z}^{2} \cdot y, -4, {x}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, {x}^{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, {x}^{2}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if 2.00000000000000003e151 < z

   1. Initial program 77.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot x - \color{blue}{t \cdot \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

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

         \[\leadsto x \cdot x - \left(-4 \cdot y - \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      4. metadata-eval N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y - -4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      5. distribute-lft-out-- N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      11. pow2 N/A

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

      12. lift-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 94.0

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

   10. Applied rewrites 94.0%

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

Alternative 4: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z\_m \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\left(-4 \cdot y\right) \cdot \mathsf{fma}\left(z\_m, z\_m, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m \cdot y\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 8.2e+37)
   (fma (* 4.0 t) y (* x x))
   (if (<= z_m 5e+126)
     (* (* -4.0 y) (fma z_m z_m (- t)))
     (* (* (* z_m y) z_m) -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 8.2e+37) {
		tmp = fma((4.0 * t), y, (x * x));
	} else if (z_m <= 5e+126) {
		tmp = (-4.0 * y) * fma(z_m, z_m, -t);
	} else {
		tmp = ((z_m * y) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 8.2e+37)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	elseif (z_m <= 5e+126)
		tmp = Float64(Float64(-4.0 * y) * fma(z_m, z_m, Float64(-t)));
	else
		tmp = Float64(Float64(Float64(z_m * y) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 8.2e+37], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 5e+126], N[(N[(-4.0 * y), $MachinePrecision] * N[(z$95$m * z$95$m + (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 8.1999999999999996e37

   1. Initial program 93.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + 4 \cdot \left(\color{blue}{t} \cdot y\right) \]

      3. +-commutative N/A

         \[\leadsto 4 \cdot \left(t \cdot y\right) + \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot y\right) \cdot 4 + {\color{blue}{x}}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, \color{blue}{4}, {x}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

         \[\leadsto \left(4 \cdot t\right) \cdot y + {x}^{\color{blue}{2}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot t, \color{blue}{y}, {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot t, y, {x}^{2}\right) \]

      9. pow2 N/A

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

      10. lift-*.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if 8.1999999999999996e37 < z < 4.99999999999999977e126

   1. Initial program 99.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-4 \cdot y\right) \cdot \color{blue}{\left({z}^{2} - t\right)} \]

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \left(-4 \cdot y\right) \cdot \left({z}^{2} - t\right) \]

      6. *-lft-identity N/A

         \[\leadsto \left(-4 \cdot y\right) \cdot \left({z}^{2} - 1 \cdot \color{blue}{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(-4 \cdot y\right) \cdot \left({z}^{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot t\right) \]

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

         \[\leadsto \left(-4 \cdot y\right) \cdot \left({z}^{2} + \color{blue}{-1 \cdot t}\right) \]

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \left(-4 \cdot y\right) \cdot \mathsf{fma}\left(z, \color{blue}{z}, -1 \cdot t\right) \]

      11. mul-1-neg N/A

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

      12. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if 4.99999999999999977e126 < z

   1. Initial program 79.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot x - \color{blue}{t \cdot \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

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

         \[\leadsto x \cdot x - \left(-4 \cdot y - \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      4. metadata-eval N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y - -4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      5. distribute-lft-out-- N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      11. pow2 N/A

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

      12. lift-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

Alternative 5: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z\_m, z\_m, -t\right), -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m \cdot y\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2e+151)
   (fma (fma z_m z_m (- t)) (* -4.0 y) (* x x))
   (* (* (* z_m y) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e+151) {
		tmp = fma(fma(z_m, z_m, -t), (-4.0 * y), (x * x));
	} else {
		tmp = ((z_m * y) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2e+151)
		tmp = fma(fma(z_m, z_m, Float64(-t)), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z_m * y) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 2e+151], N[(N[(z$95$m * z$95$m + (-t)), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000003e151

   1. Initial program 93.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

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

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) + {x}^{2}} \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot \left({z}^{2} - t\right)} + {x}^{2} \]

      15. pow2 N/A

          \[\leadsto \left(-4 \cdot y\right) \cdot \left(\color{blue}{z \cdot z} - t\right) + {x}^{2} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)} + {x}^{2} \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z - t, -4 \cdot y, {x}^{2}\right)} \]

   4. Applied rewrites 95.6%

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

   if 2.00000000000000003e151 < z

   1. Initial program 77.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot x - \color{blue}{t \cdot \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

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

         \[\leadsto x \cdot x - \left(-4 \cdot y - \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      4. metadata-eval N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y - -4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      5. distribute-lft-out-- N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      11. pow2 N/A

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

      12. lift-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 94.0

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

   10. Applied rewrites 94.0%

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

Alternative 6: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m \cdot y\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1.3e+38) (fma (* 4.0 t) y (* x x)) (* (* (* z_m y) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.3e+38) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = ((z_m * y) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.3e+38)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z_m * y) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 1.3e+38], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.3e38

   1. Initial program 93.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + 4 \cdot \left(\color{blue}{t} \cdot y\right) \]

      3. +-commutative N/A

         \[\leadsto 4 \cdot \left(t \cdot y\right) + \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot y\right) \cdot 4 + {\color{blue}{x}}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, \color{blue}{4}, {x}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

         \[\leadsto \left(4 \cdot t\right) \cdot y + {x}^{\color{blue}{2}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot t, \color{blue}{y}, {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot t, y, {x}^{2}\right) \]

      9. pow2 N/A

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

      10. lift-*.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if 1.3e38 < z

   1. Initial program 86.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot x - \color{blue}{t \cdot \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y + 4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot \color{blue}{t} \]

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

         \[\leadsto x \cdot x - \left(-4 \cdot y - \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      4. metadata-eval N/A

         \[\leadsto x \cdot x - \left(-4 \cdot y - -4 \cdot \frac{y \cdot {z}^{2}}{t}\right) \cdot t \]

      5. distribute-lft-out-- N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{y \cdot {z}^{2}}{t}\right)\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot x - \left(-4 \cdot \left(y - \frac{{z}^{2} \cdot y}{t}\right)\right) \cdot t \]

      11. pow2 N/A

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

      12. lift-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 85.8

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

   10. Applied rewrites 85.8%

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

Alternative 7: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3700000:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= x 3700000.0) (* (* 4.0 t) y) (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 3700000.0) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

z_m =     private
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_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 3700000.0d0) then
        tmp = (4.0d0 * t) * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 3700000.0) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if x <= 3700000.0:
		tmp = (4.0 * t) * y
	else:
		tmp = x * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (x <= 3700000.0)
		tmp = Float64(Float64(4.0 * t) * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (x <= 3700000.0)
		tmp = (4.0 * t) * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[x, 3700000.0], N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7e6

   1. Initial program 92.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto x \cdot \color{blue}{x} \]

      2. lift-*.f64 29.0

         \[\leadsto x \cdot \color{blue}{x} \]

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{x \cdot x} \]

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 37.3

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

   8. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 37.3

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

   10. Applied rewrites 37.3%

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

   if 3.7e6 < x

   1. Initial program 89.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto x \cdot \color{blue}{x} \]

      2. lift-*.f64 78.4

         \[\leadsto x \cdot \color{blue}{x} \]

   5. Applied rewrites 78.4%

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

Alternative 8: 41.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \cdot x \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t) :precision binary64 (* x x))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	return x * x;
}

Fortran

z_m =     private
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_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = x * x
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return x * x;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return x * x

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	return Float64(x * x)
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{2}} \]
4. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto x \cdot \color{blue}{x} \]

   2. lift-*.f64 40.2

      \[\leadsto x \cdot \color{blue}{x} \]

5. Applied rewrites 40.2%

   \[\leadsto \color{blue}{x \cdot x} \]
6. Add Preprocessing

Developer Target 1: 91.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((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) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025057 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))