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

Percentage Accurate: 97.8% → 98.9%

Time: 4.7s

Alternatives: 14

Speedup: 0.5×

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, a, b, c)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, a, b, c)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c)))
   (if (<= t_1 INFINITY) t_1 (fma (* 0.0625 t) z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma((0.0625 * t), z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) c) < +inf.0

   1. Initial program 99.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) c)

   1. Initial program 0.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x\right) \]

       2. lift-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ t_2 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-61} \lor \neg \left(t\_2 \leq 500\right):\\ \;\;\;\;\left(\frac{t\_1}{a} - 0.25 \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 t) z (fma y x c))) (t_2 (/ (* a b) 4.0)))
   (if (or (<= t_2 -2e-61) (not (<= t_2 500.0)))
     (* (- (/ t_1 a) (* 0.25 b)) a)
     t_1)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((0.0625 * t), z, fma(y, x, c));
	double t_2 = (a * b) / 4.0;
	double tmp;
	if ((t_2 <= -2e-61) || !(t_2 <= 500.0)) {
		tmp = ((t_1 / a) - (0.25 * b)) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * t), z, fma(y, x, c))
	t_2 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_2 <= -2e-61) || !(t_2 <= 500.0))
		tmp = Float64(Float64(Float64(t_1 / a) - Float64(0.25 * b)) * a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -2e-61], N[Not[LessEqual[t$95$2, 500.0]], $MachinePrecision]], N[(N[(N[(t$95$1 / a), $MachinePrecision] - N[(0.25 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.0000000000000001e-61 or 500 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   if -2.0000000000000001e-61 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 500

   1. Initial program 98.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -2 \cdot 10^{-61} \lor \neg \left(\frac{a \cdot b}{4} \leq 500\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-36}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* 0.0625 t) z c)))
   (if (<= t_1 -4e+43)
     t_2
     (if (<= t_1 -1e-36)
       (* -0.25 (* b a))
       (if (<= t_1 2e+230) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((0.0625 * t), z, c);
	double tmp;
	if (t_1 <= -4e+43) {
		tmp = t_2;
	} else if (t_1 <= -1e-36) {
		tmp = -0.25 * (b * a);
	} else if (t_1 <= 2e+230) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(0.0625 * t), z, c)
	tmp = 0.0
	if (t_1 <= -4e+43)
		tmp = t_2;
	elseif (t_1 <= -1e-36)
		tmp = Float64(-0.25 * Float64(b * a));
	elseif (t_1 <= 2e+230)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+43], t$95$2, If[LessEqual[t$95$1, -1e-36], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+230], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.00000000000000006e43 or 2.0000000000000002e230 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 94.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 90.2

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, c\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 82.0%

       \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, c\right) \]

   if -4.00000000000000006e43 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -9.9999999999999994e-37

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 65.1

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -9.9999999999999994e-37 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000002e230

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 70.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 61.5

          \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   11. Applied rewrites 61.5%

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

Alternative 4: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+163}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* x y) (/ (* z t) 16.0))))
   (if (or (<= t_1 -5e+132) (not (<= t_1 5e+163)))
     (fma (* 0.0625 t) z (* y x))
     (+ (* -0.25 (* b a)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if ((t_1 <= -5e+132) || !(t_1 <= 5e+163)) {
		tmp = fma((0.0625 * t), z, (y * x));
	} else {
		tmp = (-0.25 * (b * a)) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if ((t_1 <= -5e+132) || !(t_1 <= 5e+163))
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	else
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+132], N[Not[LessEqual[t$95$1, 5e+163]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5.0000000000000001e132 or 5e163 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 93.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x\right) \]

       2. lift-*.f64 83.9

          \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right) \]

   11. Applied rewrites 83.9%

       \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right) \]

   if -5.0000000000000001e132 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5e163

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) + c \]

      3. lower-*.f64 73.4

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) + c \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{z \cdot t}{16} \leq -5 \cdot 10^{+132} \lor \neg \left(x \cdot y + \frac{z \cdot t}{16} \leq 5 \cdot 10^{+163}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \left(0.0625 \cdot t\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-36}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* (* 0.0625 t) z)))
   (if (<= t_1 -5e+132)
     t_2
     (if (<= t_1 -1e-36)
       (* -0.25 (* b a))
       (if (<= t_1 1e+236) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = (0.0625 * t) * z;
	double tmp;
	if (t_1 <= -5e+132) {
		tmp = t_2;
	} else if (t_1 <= -1e-36) {
		tmp = -0.25 * (b * a);
	} else if (t_1 <= 1e+236) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(Float64(0.0625 * t) * z)
	tmp = 0.0
	if (t_1 <= -5e+132)
		tmp = t_2;
	elseif (t_1 <= -1e-36)
		tmp = Float64(-0.25 * Float64(b * a));
	elseif (t_1 <= 1e+236)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0625 * t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+132], t$95$2, If[LessEqual[t$95$1, -1e-36], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+236], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.0000000000000001e132 or 1.00000000000000005e236 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      3. lower-*.f64 79.6

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 \]

   5. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot \color{blue}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot \color{blue}{z} \]

      6. lift-*.f64 80.8

         \[\leadsto \left(0.0625 \cdot t\right) \cdot z \]

   7. Applied rewrites 80.8%

      \[\leadsto \left(0.0625 \cdot t\right) \cdot \color{blue}{z} \]

   if -5.0000000000000001e132 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -9.9999999999999994e-37

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 55.4

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -9.9999999999999994e-37 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.00000000000000005e236

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 70.2

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 61.7

          \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   11. Applied rewrites 61.7%

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

Alternative 6: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -5000.0)
     (+ (fma (* -0.25 b) a (* (* t z) 0.0625)) c)
     (if (<= t_1 2e+29)
       (fma (* 0.0625 t) z (fma y x c))
       (fma (* -0.25 b) a (fma (* t z) 0.0625 (* y x)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = fma((-0.25 * b), a, ((t * z) * 0.0625)) + c;
	} else if (t_1 <= 2e+29) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = fma((-0.25 * b), a, fma((t * z), 0.0625, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(fma(Float64(-0.25 * b), a, Float64(Float64(t * z) * 0.0625)) + c);
	elseif (t_1 <= 2e+29)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = fma(Float64(-0.25 * b), a, fma(Float64(t * z), 0.0625, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+29], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5e3

   1. Initial program 96.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

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

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      8. lower-*.f64 85.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\frac{-1}{4} \cdot \left(b \cdot a\right)}\right) + c \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\frac{-1}{4}} \cdot \left(b \cdot a\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) + c \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{\frac{1}{16}} \cdot \left(t \cdot z\right)\right) + c \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + \color{blue}{\frac{1}{16}} \cdot \left(t \cdot z\right)\right) + c \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16}\right) + c \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16}\right) + c \]

      12. lower-*.f64 85.4

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c \]

   7. Applied rewrites 85.4%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, \left(t \cdot z\right) \cdot 0.0625\right) + c \]

   if -5e3 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.99999999999999983e29

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 99.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   if 1.99999999999999983e29 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

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

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      8. lower-*.f64 84.6

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\frac{-1}{4} \cdot \left(b \cdot a\right)}\right) + c \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\frac{-1}{4}} \cdot \left(b \cdot a\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) + c \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{\frac{1}{16}} \cdot \left(t \cdot z\right)\right) + c \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + \color{blue}{\frac{1}{16}} \cdot \left(t \cdot z\right)\right) + c \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16}\right) + c \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16}\right) + c \]

      12. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c \]

   7. Applied rewrites 84.6%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, \left(t \cdot z\right) \cdot 0.0625\right) + c \]

   8. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot b\right) + \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot a\right) + \left(\frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)} + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{4} \cdot b\right) \cdot a + \left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, \frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16} + x \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \mathsf{fma}\left(t \cdot z, \frac{1}{16}, x \cdot y\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \mathsf{fma}\left(t \cdot z, \frac{1}{16}, x \cdot y\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \mathsf{fma}\left(t \cdot z, \frac{1}{16}, y \cdot x\right)\right) \]

      12. lift-*.f64 93.1

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\right) \]

   10. Applied rewrites 93.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-19} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e-19) (not (<= t_1 2e+20)))
     (+ (fma y x (* -0.25 (* b a))) c)
     (fma (* 0.0625 t) z (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e-19) || !(t_1 <= 2e+20)) {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	} else {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -5e-19) || !(t_1 <= 2e+20))
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	else
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-19], N[Not[LessEqual[t$95$1, 2e+20]], $MachinePrecision]], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000004e-19 or 2e20 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

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

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 85.2

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]

   if -5.0000000000000004e-19 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e20

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 99.6

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{-19} \lor \neg \left(\frac{a \cdot b}{4} \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -5000.0)
     (+ (fma (* -0.25 b) a (* (* t z) 0.0625)) c)
     (if (<= t_1 2e+20)
       (fma (* 0.0625 t) z (fma y x c))
       (+ (fma y x (* -0.25 (* b a))) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = fma((-0.25 * b), a, ((t * z) * 0.0625)) + c;
	} else if (t_1 <= 2e+20) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(fma(Float64(-0.25 * b), a, Float64(Float64(t * z) * 0.0625)) + c);
	elseif (t_1 <= 2e+20)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+20], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5e3

   1. Initial program 96.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

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

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      8. lower-*.f64 85.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\frac{-1}{4} \cdot \left(b \cdot a\right)}\right) + c \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\frac{-1}{4}} \cdot \left(b \cdot a\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) + c \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{\frac{1}{16}} \cdot \left(t \cdot z\right)\right) + c \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + \color{blue}{\frac{1}{16}} \cdot \left(t \cdot z\right)\right) + c \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \frac{1}{16} \cdot \left(t \cdot z\right)\right) + c \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16}\right) + c \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \left(t \cdot z\right) \cdot \frac{1}{16}\right) + c \]

      12. lower-*.f64 85.4

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c \]

   7. Applied rewrites 85.4%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, \left(t \cdot z\right) \cdot 0.0625\right) + c \]

   if -5e3 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e20

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 98.9

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   if 2e20 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

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

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 89.1

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+167} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+115}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+167) (not (<= t_1 2e+115)))
     (+ (* -0.25 (* b a)) c)
     (fma (* 0.0625 t) z (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e+167) || !(t_1 <= 2e+115)) {
		tmp = (-0.25 * (b * a)) + c;
	} else {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -5e+167) || !(t_1 <= 2e+115))
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	else
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+167], N[Not[LessEqual[t$95$1, 2e+115]], $MachinePrecision]], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.9999999999999997e167 or 2e115 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) + c \]

      3. lower-*.f64 80.9

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) + c \]

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} + c \]

   if -4.9999999999999997e167 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e115

   1. Initial program 98.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 91.5

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+167} \lor \neg \left(\frac{a \cdot b}{4} \leq 2 \cdot 10^{+115}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-19} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e-19) (not (<= t_1 2e+20)))
     (+ (* -0.25 (* b a)) c)
     (fma (* 0.0625 t) z c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e-19) || !(t_1 <= 2e+20)) {
		tmp = (-0.25 * (b * a)) + c;
	} else {
		tmp = fma((0.0625 * t), z, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -5e-19) || !(t_1 <= 2e+20))
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	else
		tmp = fma(Float64(0.0625 * t), z, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-19], N[Not[LessEqual[t$95$1, 2e+20]], $MachinePrecision]], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000004e-19 or 2e20 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) + c \]

      3. lower-*.f64 71.8

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) + c \]

   5. Applied rewrites 71.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} + c \]

   if -5.0000000000000004e-19 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e20

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 99.6

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, c\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 69.2%

       \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, c\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{-19} \lor \neg \left(\frac{a \cdot b}{4} \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+96} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+110}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+96) (not (<= t_1 5e+110)))
     (* -0.25 (* b a))
     (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e+96) || !(t_1 <= 5e+110)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -5e+96) || !(t_1 <= 5e+110))
		tmp = Float64(-0.25 * Float64(b * a));
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+96], N[Not[LessEqual[t$95$1, 5e+110]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000004e96 or 4.99999999999999978e110 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 67.0

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -5.0000000000000004e96 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.99999999999999978e110

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 93.9

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   8. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 55.4

          \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   11. Applied rewrites 55.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+96} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+110}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+74} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+106}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -2e+74) (not (<= (* x y) 5e+106))) (* y x) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -2e+74) || !((x * y) <= 5e+106)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	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, a, b, c)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((x * y) <= (-2d+74)) .or. (.not. ((x * y) <= 5d+106))) then
        tmp = y * x
    else
        tmp = c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -2e+74) || !((x * y) <= 5e+106)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -2e+74) or not ((x * y) <= 5e+106):
		tmp = y * x
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+74) || !(Float64(x * y) <= 5e+106))
		tmp = Float64(y * x);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((x * y) <= -2e+74) || ~(((x * y) <= 5e+106)))
		tmp = y * x;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+74], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+106]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.9999999999999999e74 or 4.9999999999999998e106 < (*.f64 x y)

   1. Initial program 94.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -1.9999999999999999e74 < (*.f64 x y) < 4.9999999999999998e106

   1. Initial program 99.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 26.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+74} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+106}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.2% accurate, 6.7× speedup

Math

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

FPCore

(FPCore (x y z t a b c) :precision binary64 (fma y x c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(y, x, c);
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(y, x, c)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(y * x + c), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

5. Applied rewrites 86.1%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

   4. associate-+r+ N/A

      \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

   7. lift-fma.f64 74.7

      \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

8. Applied rewrites 74.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

9. Taylor expanded in z around 0

   \[\leadsto c + \color{blue}{x \cdot y} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto c + y \cdot x \]

    2. +-commutative N/A

       \[\leadsto y \cdot x + c \]

    3. lift-fma.f64 44.2

       \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

11. Applied rewrites 44.2%

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
12. Add Preprocessing

Alternative 14: 22.4% accurate, 47.0× speedup

Math

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

FPCore

(FPCore (x y z t a b c) :precision binary64 c)

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return c;
}

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, a, b, c)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c
end function

Java

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

Python

def code(x, y, z, t, a, b, c):
	return c

Julia

function code(x, y, z, t, a, b, c)
	return c
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := c

Derivation

1. Initial program 97.7%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

3. Taylor expanded in c around inf

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

5. Applied rewrites 21.5%

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

Reproduce

herbie shell --seed 2025064 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))