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

Percentage Accurate: 90.6% → 97.5%

Time: 3.4s

Alternatives: 9

Speedup: 1.0×

• 48.3% 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: 90.6% 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: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(t \cdot y - \left(z \cdot y\right) \cdot z\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{t}, z, -1\right), -4 \cdot \left(y \cdot t\right), x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2e+69)
   (fma x x (* (- (* t y) (* (* z y) z)) 4.0))
   (fma (fma (/ z t) z -1.0) (* -4.0 (* y t)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2e+69) {
		tmp = fma(x, x, (((t * y) - ((z * y) * z)) * 4.0));
	} else {
		tmp = fma(fma((z / t), z, -1.0), (-4.0 * (y * t)), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2e+69)
		tmp = fma(x, x, Float64(Float64(Float64(t * y) - Float64(Float64(z * y) * z)) * 4.0));
	else
		tmp = fma(fma(Float64(z / t), z, -1.0), Float64(-4.0 * Float64(y * t)), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2e+69], N[(x * x + N[(N[(N[(t * y), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * z + -1.0), $MachinePrecision] * N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e69

   1. Initial program 90.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 92.7

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

   3. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 95.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

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

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if 2.0000000000000001e69 < t

   1. Initial program 90.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 92.7

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

   3. Applied rewrites 92.7%

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{t}, z, -1\right), -4 \cdot \left(y \cdot t\right), x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(t \cdot y - \left(z \cdot y\right) \cdot z\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{t}, z, -1\right) \cdot \left(-4 \cdot y\right), t, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.1e+69)
   (fma x x (* (- (* t y) (* (* z y) z)) 4.0))
   (fma (* (fma (/ z t) z -1.0) (* -4.0 y)) t (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.1e+69) {
		tmp = fma(x, x, (((t * y) - ((z * y) * z)) * 4.0));
	} else {
		tmp = fma((fma((z / t), z, -1.0) * (-4.0 * y)), t, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.1e+69)
		tmp = fma(x, x, Float64(Float64(Float64(t * y) - Float64(Float64(z * y) * z)) * 4.0));
	else
		tmp = fma(Float64(fma(Float64(z / t), z, -1.0) * Float64(-4.0 * y)), t, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.1e+69], N[(x * x + N[(N[(N[(t * y), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / t), $MachinePrecision] * z + -1.0), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision] * t + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.10000000000000015e69

   1. Initial program 90.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 92.7

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

   3. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 95.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

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

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if 2.10000000000000015e69 < t

   1. Initial program 90.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 92.7

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

   3. Applied rewrites 92.7%

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

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{t}, z, -1\right) \cdot \left(-4 \cdot y\right), t, x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 92.7

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

3. Applied rewrites 92.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 95.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 95.4

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

5. Applied rewrites 95.4%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lower--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

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

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

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

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

   12. remove-double-neg N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 95.1

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

7. Applied rewrites 95.1%

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

Alternative 4: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 92.7

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

3. Applied rewrites 92.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

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

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 92.7

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

5. Applied rewrites 92.7%

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

Alternative 5: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 92.7

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

3. Applied rewrites 92.7%

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

Alternative 6: 66.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in z around 0

   \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(-1 \cdot t\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 65.7

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

4. Applied rewrites 65.7%

   \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(-1 \cdot t\right)} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 66.8

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

   14. lift-*.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lift-neg.f64 66.8

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

6. Applied rewrites 66.8%

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

7. Taylor expanded in z around 0

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

   1. lower-*.f64 66.8

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

9. Applied rewrites 66.8%

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

Alternative 7: 66.3% 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 90.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 92.7

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

3. Applied rewrites 92.7%

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

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 66.3%

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

Alternative 8: 32.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 32.1

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

4. Applied rewrites 32.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 32.1

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 32.1

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

6. Applied rewrites 32.1%

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

Alternative 9: 32.1% 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 90.6%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 32.1

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

4. Applied rewrites 32.1%

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

Reproduce

herbie shell --seed 2025159 
(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))))