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

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 10

Speedup: 2.2×

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, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   2. lower-fma.f64 62.8

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 100.0%

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

Alternative 2: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+96} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-31}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -5e+96) (not (<= t_1 5e-31)))
     (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 <= -5e+96) || !(t_1 <= 5e-31)) {
		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 <= -5e+96) || !(t_1 <= 5e-31))
		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, -5e+96], N[Not[LessEqual[t$95$1, 5e-31]], $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)) < -5.0000000000000004e96 or 5e-31 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.4%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if -5.0000000000000004e96 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 5e-31

   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. +-commutative N/A

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

      2. lower-fma.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -5 \cdot 10^{+96} \lor \neg \left(\frac{y \cdot z}{2} \leq 5 \cdot 10^{-31}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (<= t_1 -5e+96)
     (fma -0.5 (* z y) t)
     (if (<= t_1 5e-31) (fma 0.125 x t) (fma (* -0.5 z) y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if (t_1 <= -5e+96) {
		tmp = fma(-0.5, (z * y), t);
	} else if (t_1 <= 5e-31) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = fma((-0.5 * z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if (t_1 <= -5e+96)
		tmp = fma(-0.5, Float64(z * y), t);
	elseif (t_1 <= 5e-31)
		tmp = fma(0.125, x, t);
	else
		tmp = fma(Float64(-0.5 * z), y, t);
	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, -5e+96], N[(-0.5 * N[(z * y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, 5e-31], N[(0.125 * x + t), $MachinePrecision], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -5.0000000000000004e96

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -5.0000000000000004e96 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 5e-31

   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. +-commutative N/A

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

      2. lower-fma.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 5e-31 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.0%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 80.7%

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

Alternative 4: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+110} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+110}\right):\\ \;\;\;\;-0.5 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -5e+110) (not (<= t_1 5e+110)))
     (* -0.5 (* y z))
     (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 <= -5e+110) || !(t_1 <= 5e+110)) {
		tmp = -0.5 * (y * z);
	} 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 <= -5e+110) || !(t_1 <= 5e+110))
		tmp = Float64(-0.5 * Float64(y * z));
	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, -5e+110], N[Not[LessEqual[t$95$1, 5e+110]], $MachinePrecision]], N[(-0.5 * N[(y * z), $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)) < -4.99999999999999978e110 or 4.99999999999999978e110 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.1%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, \mathsf{fma}\left(z, \frac{y}{-2}, t\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 82.7

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

   10. Applied rewrites 82.7%

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

   if -4.99999999999999978e110 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 4.99999999999999978e110

   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. +-commutative N/A

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

      2. lower-fma.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -5 \cdot 10^{+110} \lor \neg \left(\frac{y \cdot z}{2} \leq 5 \cdot 10^{+110}\right):\\ \;\;\;\;-0.5 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+110}:\\ \;\;\;\;-0.5 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (<= t_1 -5e+110)
     (* -0.5 (* y z))
     (if (<= t_1 5e+110) (fma 0.125 x t) (* (* z -0.5) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if (t_1 <= -5e+110) {
		tmp = -0.5 * (y * z);
	} else if (t_1 <= 5e+110) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = (z * -0.5) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if (t_1 <= -5e+110)
		tmp = Float64(-0.5 * Float64(y * z));
	elseif (t_1 <= 5e+110)
		tmp = fma(0.125, x, t);
	else
		tmp = Float64(Float64(z * -0.5) * y);
	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, -5e+110], N[(-0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+110], N[(0.125 * x + t), $MachinePrecision], N[(N[(z * -0.5), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -4.99999999999999978e110

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, \mathsf{fma}\left(z, \frac{y}{-2}, t\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 8.4

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

   if -4.99999999999999978e110 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 4.99999999999999978e110

   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. +-commutative N/A

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

      2. lower-fma.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if 4.99999999999999978e110 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 98.1%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac2 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, \mathsf{fma}\left(z, \frac{y}{-2}, t\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 12.6

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 77.5

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

   10. Applied rewrites 77.5%

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

   12. Applied rewrites 79.4%

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

Alternative 6: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+30} \lor \neg \left(x \leq 9.5 \cdot 10^{+116}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.55e+30) (not (<= x 9.5e+116)))
   (fma -0.5 (* y z) (* 0.125 x))
   (fma (* -0.5 z) y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e+30) || !(x <= 9.5e+116)) {
		tmp = fma(-0.5, (y * z), (0.125 * x));
	} else {
		tmp = fma((-0.5 * z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.55e+30) || !(x <= 9.5e+116))
		tmp = fma(-0.5, Float64(y * z), Float64(0.125 * x));
	else
		tmp = fma(Float64(-0.5 * z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.55e+30], N[Not[LessEqual[x, 9.5e+116]], $MachinePrecision]], N[(-0.5 * N[(y * z), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5499999999999999e30 or 9.5000000000000004e116 < 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 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.2

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

   5. Applied rewrites 38.2%

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 85.8

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

   10. Applied rewrites 85.8%

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

   if -1.5499999999999999e30 < x < 9.5000000000000004e116

   1. Initial program 99.5%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 93.2%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot z, \color{blue}{y}, t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+30} \lor \neg \left(x \leq 9.5 \cdot 10^{+116}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -0.5, y, 0.125 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.55e+30)
   (fma -0.5 (* y z) (* 0.125 x))
   (if (<= x 9.5e+116) (fma (* -0.5 z) y t) (fma (* z -0.5) y (* 0.125 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.55e+30) {
		tmp = fma(-0.5, (y * z), (0.125 * x));
	} else if (x <= 9.5e+116) {
		tmp = fma((-0.5 * z), y, t);
	} else {
		tmp = fma((z * -0.5), y, (0.125 * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.55e+30)
		tmp = fma(-0.5, Float64(y * z), Float64(0.125 * x));
	elseif (x <= 9.5e+116)
		tmp = fma(Float64(-0.5 * z), y, t);
	else
		tmp = fma(Float64(z * -0.5), y, Float64(0.125 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.55e+30], N[(-0.5 * N[(y * z), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+116], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(z * -0.5), $MachinePrecision] * y + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5499999999999999e30

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.1

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

   5. Applied rewrites 38.1%

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

   7. Applied rewrites 38.1%

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

   8. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 86.0

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

   10. Applied rewrites 86.0%

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

   if -1.5499999999999999e30 < x < 9.5000000000000004e116

   1. Initial program 99.5%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 93.2%

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

   if 9.5000000000000004e116 < 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 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 38.3%

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

   8. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 85.6

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

   10. Applied rewrites 85.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 85.6%

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

Alternative 8: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 9: 64.5% 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 99.7%

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

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

   2. lower-fma.f64 62.8

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

5. Applied rewrites 62.8%

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

Alternative 10: 32.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot x \end{array} \]

FPCore

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

C

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

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 = 0.125d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.125 * x;
}

Python

def code(x, y, z, t):
	return 0.125 * x

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.125 * x;
end

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing
3. 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. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

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

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

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

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

   8. associate-/l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-+l+ N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-/.f64 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-/l* N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, \mathsf{fma}\left(z, \frac{y}{-2}, t\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. lower-*.f64 30.7

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

7. Applied rewrites 30.7%

   \[\leadsto \color{blue}{0.125 \cdot x} \]
8. 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 2024359 
(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))