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

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 8

Speedup: 1.0×

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: 100.0% 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, 1.0× 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+24} \lor \neg \left(t\_1 \leq 10^{+72}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -2e+24) (not (<= t_1 1e+72)))
     (fma -0.5 (* z y) (* 0.125 x))
     (fma 0.125 x t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if ((t_1 <= -2e+24) || !(t_1 <= 1e+72)) {
		tmp = fma(-0.5, (z * y), (0.125 * x));
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if ((t_1 <= -2e+24) || !(t_1 <= 1e+72))
		tmp = fma(-0.5, Float64(z * y), Float64(0.125 * x));
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+24], N[Not[LessEqual[t$95$1, 1e+72]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -2e24 or 9.99999999999999944e71 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 87.7

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

   5. Applied rewrites 87.7%

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

   if -2e24 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 9.99999999999999944e71

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -2 \cdot 10^{+24} \lor \neg \left(\frac{y \cdot z}{2} \leq 10^{+72}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+24} \lor \neg \left(t\_1 \leq 10^{+85}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -2e+24) (not (<= t_1 1e+85)))
     (fma -0.5 (* z y) t)
     (fma 0.125 x t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if ((t_1 <= -2e+24) || !(t_1 <= 1e+85)) {
		tmp = fma(-0.5, (z * y), t);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if ((t_1 <= -2e+24) || !(t_1 <= 1e+85))
		tmp = fma(-0.5, Float64(z * y), t);
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+24], N[Not[LessEqual[t$95$1, 1e+85]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision] + t), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -2e24 or 1e85 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -2e24 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 1e85

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -2 \cdot 10^{+24} \lor \neg \left(\frac{y \cdot z}{2} \leq 10^{+85}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+118} \lor \neg \left(t\_1 \leq 10^{+145}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -1e+118) (not (<= t_1 1e+145)))
     (* -0.5 (* z y))
     (fma 0.125 x t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if ((t_1 <= -1e+118) || !(t_1 <= 1e+145)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if ((t_1 <= -1e+118) || !(t_1 <= 1e+145))
		tmp = Float64(-0.5 * Float64(z * y));
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+118], N[Not[LessEqual[t$95$1, 1e+145]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -9.99999999999999967e117 or 9.9999999999999999e144 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -9.99999999999999967e117 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 9.9999999999999999e144

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -1 \cdot 10^{+118} \lor \neg \left(\frac{y \cdot z}{2} \leq 10^{+145}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-81} \lor \neg \left(z \leq 4.4 \cdot 10^{-78}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, \frac{\mathsf{fma}\left(0.125, x, t\right)}{z}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.6e-81) (not (<= z 4.4e-78)))
   (* (fma -0.5 y (/ (fma 0.125 x t) z)) z)
   (fma 0.125 x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.6e-81) || !(z <= 4.4e-78)) {
		tmp = fma(-0.5, y, (fma(0.125, x, t) / z)) * z;
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.6e-81) || !(z <= 4.4e-78))
		tmp = Float64(fma(-0.5, y, Float64(fma(0.125, x, t) / z)) * z);
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.6e-81], N[Not[LessEqual[z, 4.4e-78]], $MachinePrecision]], N[(N[(-0.5 * y + N[(N[(0.125 * x + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5999999999999997e-81 or 4.3999999999999998e-78 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 98.8%

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

   if -7.5999999999999997e-81 < z < 4.3999999999999998e-78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval 91.4

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

   5. Applied rewrites 91.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-81} \lor \neg \left(z \leq 4.4 \cdot 10^{-78}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, \frac{\mathsf{fma}\left(0.125, x, t\right)}{z}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+46} \lor \neg \left(x \leq 41\right):\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.4e+46) (not (<= x 41.0))) (* 0.125 x) t))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e+46) || !(x <= 41.0)) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	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 <= (-2.4d+46)) .or. (.not. (x <= 41.0d0))) then
        tmp = 0.125d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e+46) || !(x <= 41.0)) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.4e+46) or not (x <= 41.0):
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.4e+46) || !(x <= 41.0))
		tmp = Float64(0.125 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.4e+46) || ~((x <= 41.0)))
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.4e+46], N[Not[LessEqual[x, 41.0]], $MachinePrecision]], N[(0.125 * x), $MachinePrecision], t]

Derivation

1. Split input into 2 regimes

2. if x < -2.40000000000000008e46 or 41 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{8} \cdot x} \]
   4. 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 60.6

         \[\leadsto 0.125 \cdot x \]

   5. Applied rewrites 60.6%

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

   if -2.40000000000000008e46 < x < 41

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+46} \lor \neg \left(x \leq 41\right):\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval 64.3

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

5. Applied rewrites 64.3%

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

Alternative 8: 32.9% accurate, 39.0× 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 100.0%

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

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Applied rewrites 32.8%

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

Developer Target 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ (/ x 8) t) (* (/ z 2) y)))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))