Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, B

Percentage Accurate: 99.9% → 100.0%

Time: 3.0s

Alternatives: 7

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + 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 = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + 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 = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z \cdot -0.5, y, \mathsf{fma}\left(x, 0.125, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (* z -0.5) y (fma x 0.125 t)))

C

double code(double x, double y, double z, double t) {
	return fma((z * -0.5), y, fma(x, 0.125, t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(z * -0.5), y, fma(x, 0.125, t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * -0.5), $MachinePrecision] * y + N[(x * 0.125 + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right)} + t \]

   3. sub-flip N/A

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)\right)} + t \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + \frac{1}{8} \cdot x\right)} + t \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + \left(\frac{1}{8} \cdot x + t\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot z}}{2}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

   8. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{2}}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{z}{2} \cdot y}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

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

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z}{2}\right)\right) \cdot y} + \left(\frac{1}{8} \cdot x + t\right) \]

   11. +-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{z}{2}\right)\right) \cdot y + \color{blue}{\left(t + \frac{1}{8} \cdot x\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{z}{2}\right), y, t + \frac{1}{8} \cdot x\right)} \]

   13. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot \frac{1}{2}}\right), y, t + \frac{1}{8} \cdot x\right) \]

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

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, y, t + \frac{1}{8} \cdot x\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(2\right)}}, y, t + \frac{1}{8} \cdot x\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \frac{1}{\mathsf{neg}\left(2\right)}}, y, t + \frac{1}{8} \cdot x\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{\color{blue}{-2}}, y, t + \frac{1}{8} \cdot x\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\frac{-1}{2}}, y, t + \frac{1}{8} \cdot x\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]

   20. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \color{blue}{\frac{1}{8} \cdot x} + t\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \color{blue}{x \cdot \frac{1}{8}} + t\right) \]

   22. lower-fma.f64 100.0

       \[\leadsto \mathsf{fma}\left(z \cdot -0.5, y, \color{blue}{\mathsf{fma}\left(x, \frac{1}{8}, t\right)}\right) \]

   23. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8}}, t\right)\right) \]

   24. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(z \cdot -0.5, y, \mathsf{fma}\left(x, \color{blue}{0.125}, t\right)\right) \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot -0.5, y, \mathsf{fma}\left(x, 0.125, t\right)\right)} \]
4. Add Preprocessing

Alternative 2: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (<= t_1 -10000.0)
     (fma (* -0.5 z) y t)
     (if (<= t_1 4e-38) (fma x 0.125 t) (fma -0.5 (* y z) (* 0.125 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = fma((-0.5 * z), y, t);
	} else if (t_1 <= 4e-38) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = fma(-0.5, (y * z), (0.125 * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = fma(Float64(-0.5 * z), y, t);
	elseif (t_1 <= 4e-38)
		tmp = fma(x, 0.125, t);
	else
		tmp = fma(-0.5, Float64(y * z), Float64(0.125 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[t$95$1, 4e-38], N[(x * 0.125 + t), $MachinePrecision], N[(-0.5 * N[(y * z), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -1e4

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t - \frac{1}{2} \cdot \left(y \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 69.7

         \[\leadsto t - 0.5 \cdot \left(y \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 69.7%

      \[\leadsto \color{blue}{t - 0.5 \cdot \left(y \cdot z\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right) \cdot y} \]

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

         \[\leadsto t + \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z\right) \cdot y \]

      8. metadata-eval N/A

         \[\leadsto t + \left(\frac{-1}{2} \cdot z\right) \cdot y \]

      9. *-commutative N/A

         \[\leadsto t + \left(z \cdot \frac{-1}{2}\right) \cdot y \]

      10. lift-*.f64 N/A

          \[\leadsto t + \left(z \cdot \frac{-1}{2}\right) \cdot y \]

      11. +-commutative N/A

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

      12. lower-fma.f64 69.8

          \[\leadsto \mathsf{fma}\left(z \cdot -0.5, \color{blue}{y}, t\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, t\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot z, y, t\right) \]

      15. lower-*.f64 69.8

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, y, t\right) \]

   6. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, \color{blue}{y}, t\right) \]

   if -1e4 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 3.9999999999999998e-38

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

      4. metadata-eval 63.9

         \[\leadsto t + 0.125 \cdot x \]

   4. Applied rewrites 63.9%

      \[\leadsto \color{blue}{t + 0.125 \cdot x} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot x + \color{blue}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot x + t \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{1}{8} + t \]

      6. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{8}}, t\right) \]

      7. metadata-eval 63.9

         \[\leadsto \mathsf{fma}\left(x, 0.125, t\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.125}, t\right) \]

   if 3.9999999999999998e-38 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right)} + t \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)\right)} + t \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + \frac{1}{8} \cdot x\right)} + t \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + \left(\frac{1}{8} \cdot x + t\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot z}}{2}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{2}}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{z}{2} \cdot y}\right)\right) + \left(\frac{1}{8} \cdot x + t\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z}{2}\right)\right) \cdot y} + \left(\frac{1}{8} \cdot x + t\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{2}\right)\right) \cdot y + \color{blue}{\left(t + \frac{1}{8} \cdot x\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{z}{2}\right), y, t + \frac{1}{8} \cdot x\right)} \]

      13. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot \frac{1}{2}}\right), y, t + \frac{1}{8} \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, y, t + \frac{1}{8} \cdot x\right) \]

      15. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(2\right)}}, y, t + \frac{1}{8} \cdot x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \frac{1}{\mathsf{neg}\left(2\right)}}, y, t + \frac{1}{8} \cdot x\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{\color{blue}{-2}}, y, t + \frac{1}{8} \cdot x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\frac{-1}{2}}, y, t + \frac{1}{8} \cdot x\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \color{blue}{\frac{1}{8} \cdot x} + t\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \color{blue}{x \cdot \frac{1}{8}} + t\right) \]

      22. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(z \cdot -0.5, y, \color{blue}{\mathsf{fma}\left(x, \frac{1}{8}, t\right)}\right) \]

      23. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8}}, t\right)\right) \]

      24. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(z \cdot -0.5, y, \mathsf{fma}\left(x, \color{blue}{0.125}, t\right)\right) \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot -0.5, y, \mathsf{fma}\left(x, 0.125, t\right)\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right) + \frac{1}{8} \cdot x} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \left(y \cdot z\right) + \frac{1}{8} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{y \cdot z}, \frac{1}{8} \cdot x\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, y \cdot \color{blue}{z}, \frac{1}{8} \cdot x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, y \cdot z, \frac{1}{8} \cdot x\right) \]

      5. metadata-eval 67.1

         \[\leadsto \mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ t_2 := \mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)) (t_2 (fma (* -0.5 z) y t)))
   (if (<= t_1 -10000.0) t_2 (if (<= t_1 4e-44) (fma x 0.125 t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double t_2 = fma((-0.5 * z), y, t);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 4e-44) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	t_2 = fma(Float64(-0.5 * z), y, t)
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 4e-44)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$2, If[LessEqual[t$95$1, 4e-44], N[(x * 0.125 + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -1e4 or 3.99999999999999981e-44 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t - \frac{1}{2} \cdot \left(y \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 69.7

         \[\leadsto t - 0.5 \cdot \left(y \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 69.7%

      \[\leadsto \color{blue}{t - 0.5 \cdot \left(y \cdot z\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right) \cdot y} \]

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

         \[\leadsto t + \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z\right) \cdot y \]

      8. metadata-eval N/A

         \[\leadsto t + \left(\frac{-1}{2} \cdot z\right) \cdot y \]

      9. *-commutative N/A

         \[\leadsto t + \left(z \cdot \frac{-1}{2}\right) \cdot y \]

      10. lift-*.f64 N/A

          \[\leadsto t + \left(z \cdot \frac{-1}{2}\right) \cdot y \]

      11. +-commutative N/A

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

      12. lower-fma.f64 69.8

          \[\leadsto \mathsf{fma}\left(z \cdot -0.5, \color{blue}{y}, t\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, t\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot z, y, t\right) \]

      15. lower-*.f64 69.8

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, y, t\right) \]

   6. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, \color{blue}{y}, t\right) \]

   if -1e4 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 3.99999999999999981e-44

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

      4. metadata-eval 63.9

         \[\leadsto t + 0.125 \cdot x \]

   4. Applied rewrites 63.9%

      \[\leadsto \color{blue}{t + 0.125 \cdot x} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot x + \color{blue}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot x + t \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{1}{8} + t \]

      6. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{8}}, t\right) \]

      7. metadata-eval 63.9

         \[\leadsto \mathsf{fma}\left(x, 0.125, t\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.125}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ t_2 := -0.5 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)) (t_2 (* -0.5 (* y z))))
   (if (<= t_1 -1e+185) t_2 (if (<= t_1 5e+152) (fma x 0.125 t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double t_2 = -0.5 * (y * z);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = t_2;
	} else if (t_1 <= 5e+152) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	t_2 = Float64(-0.5 * Float64(y * z))
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = t_2;
	elseif (t_1 <= 5e+152)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], t$95$2, If[LessEqual[t$95$1, 5e+152], N[(x * 0.125 + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -9.9999999999999998e184 or 5e152 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

      4. metadata-eval 63.9

         \[\leadsto t + 0.125 \cdot x \]

   4. Applied rewrites 63.9%

      \[\leadsto \color{blue}{t + 0.125 \cdot x} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 37.6

         \[\leadsto -0.5 \cdot \left(y \cdot \color{blue}{z}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} \]

   if -9.9999999999999998e184 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 5e152

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

      4. metadata-eval 63.9

         \[\leadsto t + 0.125 \cdot x \]

   4. Applied rewrites 63.9%

      \[\leadsto \color{blue}{t + 0.125 \cdot x} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot x + \color{blue}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot x + t \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{1}{8} + t \]

      6. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{8}}, t\right) \]

      7. metadata-eval 63.9

         \[\leadsto \mathsf{fma}\left(x, 0.125, t\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.125}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.125, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x 0.125 t))

C

double code(double x, double y, double z, double t) {
	return fma(x, 0.125, t);
}

Julia

function code(x, y, z, t)
	return fma(x, 0.125, t)
end

Wolfram

code[x_, y_, z_, t_] := N[(x * 0.125 + t), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto t + \frac{1}{8} \cdot x \]

   2. lower-+.f64 N/A

      \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

   3. lower-*.f64 N/A

      \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

   4. metadata-eval 63.9

      \[\leadsto t + 0.125 \cdot x \]

4. Applied rewrites 63.9%

   \[\leadsto \color{blue}{t + 0.125 \cdot x} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto t + \frac{1}{8} \cdot x \]

   2. lift-+.f64 N/A

      \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot x + \color{blue}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{8} \cdot x + t \]

   5. *-commutative N/A

      \[\leadsto x \cdot \frac{1}{8} + t \]

   6. lift-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{8}}, t\right) \]

   7. metadata-eval 63.9

      \[\leadsto \mathsf{fma}\left(x, 0.125, t\right) \]

6. Applied rewrites 63.9%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.125}, t\right) \]
7. Add Preprocessing

Alternative 6: 49.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+152}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+82}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9.2e+152) (* 0.125 x) (if (<= x 9e+82) t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.2e+152) {
		tmp = 0.125 * x;
	} else if (x <= 9e+82) {
		tmp = t;
	} else {
		tmp = 0.125 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-9.2d+152)) then
        tmp = 0.125d0 * x
    else if (x <= 9d+82) then
        tmp = t
    else
        tmp = 0.125d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.2e+152) {
		tmp = 0.125 * x;
	} else if (x <= 9e+82) {
		tmp = t;
	} else {
		tmp = 0.125 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -9.2e+152:
		tmp = 0.125 * x
	elif x <= 9e+82:
		tmp = t
	else:
		tmp = 0.125 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9.2e+152)
		tmp = Float64(0.125 * x);
	elseif (x <= 9e+82)
		tmp = t;
	else
		tmp = Float64(0.125 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -9.2e+152)
		tmp = 0.125 * x;
	elseif (x <= 9e+82)
		tmp = t;
	else
		tmp = 0.125 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9.2e+152], N[(0.125 * x), $MachinePrecision], If[LessEqual[x, 9e+82], t, N[(0.125 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999994e152 or 8.9999999999999993e82 < x

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t - \frac{1}{2} \cdot \left(y \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 69.7

         \[\leadsto t - 0.5 \cdot \left(y \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 69.7%

      \[\leadsto \color{blue}{t - 0.5 \cdot \left(y \cdot z\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right) \cdot y} \]

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

         \[\leadsto t + \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z\right) \cdot y \]

      8. metadata-eval N/A

         \[\leadsto t + \left(\frac{-1}{2} \cdot z\right) \cdot y \]

      9. *-commutative N/A

         \[\leadsto t + \left(z \cdot \frac{-1}{2}\right) \cdot y \]

      10. lift-*.f64 N/A

          \[\leadsto t + \left(z \cdot \frac{-1}{2}\right) \cdot y \]

      11. +-commutative N/A

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

      12. lower-fma.f64 69.8

          \[\leadsto \mathsf{fma}\left(z \cdot -0.5, \color{blue}{y}, t\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \frac{-1}{2}, y, t\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot z, y, t\right) \]

      15. lower-*.f64 69.8

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, y, t\right) \]

   6. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, \color{blue}{y}, t\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{8} \cdot x} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{8} \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \color{blue}{x} \]

      3. metadata-eval 31.5

         \[\leadsto 0.125 \cdot x \]

   9. Applied rewrites 31.5%

      \[\leadsto \color{blue}{0.125 \cdot x} \]

   if -9.1999999999999994e152 < x < 8.9999999999999993e82

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto t + \frac{1}{8} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

      4. metadata-eval 63.9

         \[\leadsto t + 0.125 \cdot x \]

   4. Applied rewrites 63.9%

      \[\leadsto \color{blue}{t + 0.125 \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto t \]
   6. Step-by-step derivation

   7. Applied rewrites 34.2%

      \[\leadsto t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 34.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto t + \frac{1}{8} \cdot x \]

   2. lower-+.f64 N/A

      \[\leadsto t + \color{blue}{\frac{1}{8} \cdot x} \]

   3. lower-*.f64 N/A

      \[\leadsto t + \frac{1}{8} \cdot \color{blue}{x} \]

   4. metadata-eval 63.9

      \[\leadsto t + 0.125 \cdot x \]

4. Applied rewrites 63.9%

   \[\leadsto \color{blue}{t + 0.125 \cdot x} \]

5. Taylor expanded in x around 0

   \[\leadsto t \]
6. Step-by-step derivation

7. Applied rewrites 34.2%

   \[\leadsto t \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64
  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))