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

Percentage Accurate: 90.8% → 98.5%

Time: 3.4s

Alternatives: 9

Speedup: 0.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z\right| \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left|z\right| \cdot \left|z\right| - t, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \frac{\left(\frac{t}{\left|z\right|} - \left|z\right|\right) \cdot \left(-4 \cdot y\right)}{\frac{-1}{\left|z\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs z) 5e+152)
   (fma (- (* (fabs z) (fabs z)) t) (* -4.0 y) (* x x))
   (fma x x (/ (* (- (/ t (fabs z)) (fabs z)) (* -4.0 y)) (/ -1.0 (fabs z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(z) <= 5e+152) {
		tmp = fma(((fabs(z) * fabs(z)) - t), (-4.0 * y), (x * x));
	} else {
		tmp = fma(x, x, ((((t / fabs(z)) - fabs(z)) * (-4.0 * y)) / (-1.0 / fabs(z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(z) <= 5e+152)
		tmp = fma(Float64(Float64(abs(z) * abs(z)) - t), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = fma(x, x, Float64(Float64(Float64(Float64(t / abs(z)) - abs(z)) * Float64(-4.0 * y)) / Float64(-1.0 / abs(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[z], $MachinePrecision], 5e+152], N[(N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(N[(t / N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5e152

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 91.9

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

   3. Applied rewrites 91.9%

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

   if 5e152 < z

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   5. Applied rewrites 87.4%

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

Alternative 2: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z\right| \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left|z\right| \cdot \left|z\right| - t, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, t \cdot y - \left(\left|z\right| \cdot y\right) \cdot \left|z\right|, x \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs z) 2.5e+153)
   (fma (- (* (fabs z) (fabs z)) t) (* -4.0 y) (* x x))
   (fma 4.0 (- (* t y) (* (* (fabs z) y) (fabs z))) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(z) <= 2.5e+153) {
		tmp = fma(((fabs(z) * fabs(z)) - t), (-4.0 * y), (x * x));
	} else {
		tmp = fma(4.0, ((t * y) - ((fabs(z) * y) * fabs(z))), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(z) <= 2.5e+153)
		tmp = fma(Float64(Float64(abs(z) * abs(z)) - t), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = fma(4.0, Float64(Float64(t * y) - Float64(Float64(abs(z) * y) * abs(z))), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[z], $MachinePrecision], 2.5e+153], N[(N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(t * y), $MachinePrecision] - N[(N[(N[Abs[z], $MachinePrecision] * y), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.50000000000000009e153

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 91.9

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

   3. Applied rewrites 91.9%

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

   if 2.50000000000000009e153 < z

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

      15. remove-double-neg N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. lower--.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. remove-double-neg N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*r* N/A

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

      6. remove-double-div N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. frac-2neg N/A

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

      14. distribute-frac-neg2 N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 93.4%

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

Alternative 3: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.35 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, \left|x\right|, \left(\left(t - z \cdot z\right) \cdot y\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, \left|x\right| \cdot \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs x) 1.35e+191)
   (fma (fabs x) (fabs x) (* (* (- t (* z z)) y) 4.0))
   (fma (* t 4.0) y (* (fabs x) (fabs x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(x) <= 1.35e+191) {
		tmp = fma(fabs(x), fabs(x), (((t - (z * z)) * y) * 4.0));
	} else {
		tmp = fma((t * 4.0), y, (fabs(x) * fabs(x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(x) <= 1.35e+191)
		tmp = fma(abs(x), abs(x), Float64(Float64(Float64(t - Float64(z * z)) * y) * 4.0));
	else
		tmp = fma(Float64(t * 4.0), y, Float64(abs(x) * abs(x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.35e+191], N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(t * 4.0), $MachinePrecision] * y + N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999998e191

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   if 1.34999999999999998e191 < x

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

Alternative 4: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z\right| \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left|z\right|, \left|z\right|, t\right), -4 \cdot y, x \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs z) 5.5e+24)
   (fma (* t 4.0) y (* x x))
   (fma (fma (fabs z) (fabs z) t) (* -4.0 y) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(z) <= 5.5e+24) {
		tmp = fma((t * 4.0), y, (x * x));
	} else {
		tmp = fma(fma(fabs(z), fabs(z), t), (-4.0 * y), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(z) <= 5.5e+24)
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	else
		tmp = fma(fma(abs(z), abs(z), t), Float64(-4.0 * y), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[z], $MachinePrecision], 5.5e+24], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision] + t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.5000000000000002e24

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 5.5000000000000002e24 < z

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

      15. remove-double-neg N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

   3. Applied rewrites 93.4%

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

   5. Applied rewrites 67.1%

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

Alternative 5: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z\right| \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\left|z\right|, \left|z\right|, t\right) \cdot \left(-4 \cdot y\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs z) 3.8e+63)
   (fma (* t 4.0) y (* x x))
   (fma x x (* (fma (fabs z) (fabs z) t) (* -4.0 y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(z) <= 3.8e+63) {
		tmp = fma((t * 4.0), y, (x * x));
	} else {
		tmp = fma(x, x, (fma(fabs(z), fabs(z), t) * (-4.0 * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(z) <= 3.8e+63)
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	else
		tmp = fma(x, x, Float64(fma(abs(z), abs(z), t) * Float64(-4.0 * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[z], $MachinePrecision], 3.8e+63], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision] + t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.8000000000000001e63

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 3.8000000000000001e63 < z

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   5. Applied rewrites 67.5%

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

Alternative 6: 67.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 93.0

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

3. Applied rewrites 93.0%

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

4. Taylor expanded in z around 0

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

6. Applied rewrites 67.1%

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

Alternative 7: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 20500000:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, \left|x\right|, t \cdot \left(-4 \cdot y\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (fabs x) 20500000.0)
   (* 4.0 (* t y))
   (fma (fabs x) (fabs x) (* t (* -4.0 y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (fabs(x) <= 20500000.0) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = fma(fabs(x), fabs(x), (t * (-4.0 * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (abs(x) <= 20500000.0)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = fma(abs(x), abs(x), Float64(t * Float64(-4.0 * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Abs[x], $MachinePrecision], 20500000.0], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(t * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.05e7

   1. Initial program 90.8%

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

   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 30.8

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

   4. Applied rewrites 30.8%

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

   if 2.05e7 < x

   1. Initial program 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 42.0%

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

Alternative 8: 30.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

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 30.8

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

4. Applied rewrites 30.8%

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

Alternative 9: 6.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-flip N/A

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

   7. distribute-lft-in N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. associate-*l* N/A

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

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

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

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

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

   15. remove-double-neg N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

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

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

3. Applied rewrites 93.4%

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

5. Applied rewrites 67.1%

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

6. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 6.2

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

8. Applied rewrites 6.2%

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

Reproduce

herbie shell --seed 2025171 
(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))))