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

Percentage Accurate: 97.7% → 97.7%

Time: 7.0s

Alternatives: 11

Speedup: 1.0×

• 33.4% 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: 97.7% 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: 97.7% 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]

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

Alternative 2: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma (* t z) 0.0625 c)))
   (if (<= t_1 -5e+72)
     (fma (* -0.25 a) b (fma y x (* (* t z) 0.0625)))
     (if (<= t_1 5e+85) (fma y x t_2) (fma (* -0.25 a) b t_2)))))

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 t_2 = fma((t * z), 0.0625, c);
	double tmp;
	if (t_1 <= -5e+72) {
		tmp = fma((-0.25 * a), b, fma(y, x, ((t * z) * 0.0625)));
	} else if (t_1 <= 5e+85) {
		tmp = fma(y, x, t_2);
	} else {
		tmp = fma((-0.25 * a), b, t_2);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (t_1 <= -5e+72)
		tmp = fma(Float64(-0.25 * a), b, fma(y, x, Float64(Float64(t * z) * 0.0625)));
	elseif (t_1 <= 5e+85)
		tmp = fma(y, x, t_2);
	else
		tmp = fma(Float64(-0.25 * a), b, t_2);
	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]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+72], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+85], N[(y * x + t$95$2), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.99999999999999992e72

   1. Initial program 95.3%

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

   2. Taylor expanded in c around 0

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

      1. metadata-eval N/A

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

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

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

      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. associate-*r* N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. distribute-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -4.99999999999999992e72 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.0000000000000001e85

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 93.5%

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

   if 5.0000000000000001e85 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.7%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 85.5%

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

Alternative 3: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ t_2 := \frac{a \cdot b}{4}\\ t_3 := \mathsf{fma}\left(-0.25 \cdot a, b, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-9}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 c))
        (t_2 (/ (* a b) 4.0))
        (t_3 (fma (* -0.25 a) b t_1)))
   (if (<= t_2 -4e-9) t_3 (if (<= t_2 5e+85) (fma y x t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * z), 0.0625, c);
	double t_2 = (a * b) / 4.0;
	double t_3 = fma((-0.25 * a), b, t_1);
	double tmp;
	if (t_2 <= -4e-9) {
		tmp = t_3;
	} else if (t_2 <= 5e+85) {
		tmp = fma(y, x, t_1);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * z), 0.0625, c)
	t_2 = Float64(Float64(a * b) / 4.0)
	t_3 = fma(Float64(-0.25 * a), b, t_1)
	tmp = 0.0
	if (t_2 <= -4e-9)
		tmp = t_3;
	elseif (t_2 <= 5e+85)
		tmp = fma(y, x, t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.25 * a), $MachinePrecision] * b + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-9], t$95$3, If[LessEqual[t$95$2, 5e+85], N[(y * x + t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.00000000000000025e-9 or 5.0000000000000001e85 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 82.8%

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

   if -4.00000000000000025e-9 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.0000000000000001e85

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 94.8%

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

Alternative 4: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma (* -0.25 a) b (fma y x c))))
   (if (<= t_1 -1e-24)
     t_2
     (if (<= t_1 4e+73) (fma y x (fma (* t z) 0.0625 c)) t_2))))

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 t_2 = fma((-0.25 * a), b, fma(y, x, c));
	double tmp;
	if (t_1 <= -1e-24) {
		tmp = t_2;
	} else if (t_1 <= 4e+73) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = fma(Float64(-0.25 * a), b, fma(y, x, c))
	tmp = 0.0
	if (t_1 <= -1e-24)
		tmp = t_2;
	elseif (t_1 <= 4e+73)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(-0.25 * a), $MachinePrecision] * b + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-24], t$95$2, If[LessEqual[t$95$1, 4e+73], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999924e-25 or 3.99999999999999993e73 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. remove-double-neg N/A

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

      10. remove-double-neg N/A

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

      11. *-commutative N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 81.6

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

   4. Applied rewrites 81.6%

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

   if -9.99999999999999924e-25 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 3.99999999999999993e73

   1. Initial program 99.1%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 95.3%

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

Alternative 5: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -5e+79)
     (fma (* -0.25 a) b (* y x))
     (if (<= t_1 1e+186)
       (fma y x (fma (* t z) 0.0625 c))
       (fma (* -0.25 a) b 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+79) {
		tmp = fma((-0.25 * a), b, (y * x));
	} else if (t_1 <= 1e+186) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = fma((-0.25 * a), b, 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+79)
		tmp = fma(Float64(-0.25 * a), b, Float64(y * x));
	elseif (t_1 <= 1e+186)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = fma(Float64(-0.25 * a), b, 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, -5e+79], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+186], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5e79

   1. Initial program 95.2%

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

   2. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. remove-double-neg N/A

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

      10. remove-double-neg N/A

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

      11. *-commutative N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot a, b, x \cdot y\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-*.f64 77.4

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

   7. Applied rewrites 77.4%

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

   if -5e79 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999998e185

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 90.8%

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

   if 9.9999999999999998e185 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot a, b, c\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 79.7%

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

Alternative 6: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* (* t z) 0.0625))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -5e+188)
     t_1
     (if (<= t_2 -5e+23)
       (fma (* t z) 0.0625 c)
       (if (<= t_2 5e+83) (fma (* -0.25 a) b c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, ((t * z) * 0.0625));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -5e+188) {
		tmp = t_1;
	} else if (t_2 <= -5e+23) {
		tmp = fma((t * z), 0.0625, c);
	} else if (t_2 <= 5e+83) {
		tmp = fma((-0.25 * a), b, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(Float64(t * z) * 0.0625))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -5e+188)
		tmp = t_1;
	elseif (t_2 <= -5e+23)
		tmp = fma(Float64(t * z), 0.0625, c);
	elseif (t_2 <= 5e+83)
		tmp = fma(Float64(-0.25 * a), b, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+188], t$95$1, If[LessEqual[t$95$2, -5e+23], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[t$95$2, 5e+83], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5.0000000000000001e188 or 5.00000000000000029e83 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.5%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if -5.0000000000000001e188 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -4.9999999999999999e23

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   if -4.9999999999999999e23 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.00000000000000029e83

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot a, b, c\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 83.3%

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

Alternative 7: 65.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5e+99)
   (fma y x c)
   (if (<= (* x y) 2e+88) (fma (* t z) 0.0625 c) (fma 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) <= -5e+99) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= 2e+88) {
		tmp = fma((t * z), 0.0625, c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -5e+99)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= 2e+88)
		tmp = fma(Float64(t * z), 0.0625, c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+99], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+88], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000008e99 or 1.99999999999999992e88 < (*.f64 x y)

   1. Initial program 95.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 71.3%

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

   if -5.00000000000000008e99 < (*.f64 x y) < 1.99999999999999992e88

   1. Initial program 99.1%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 61.5

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

   7. Applied rewrites 61.5%

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

Alternative 8: 62.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -2e+75) t_2 (if (<= t_1 1e+186) (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 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -2e+75) {
		tmp = t_2;
	} else if (t_1 <= 1e+186) {
		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(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -2e+75)
		tmp = t_2;
	elseif (t_1 <= 1e+186)
		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[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+75], t$95$2, If[LessEqual[t$95$1, 1e+186], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.99999999999999985e75 or 9.9999999999999998e185 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.3%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 67.9

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

   4. Applied rewrites 67.9%

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

   if -1.99999999999999985e75 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999998e185

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   3. 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. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 59.4%

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

Alternative 9: 48.4% accurate, 4.1× 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. Taylor expanded in a around 0

   \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
3. 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. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. remove-double-neg N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. remove-double-neg N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. *-commutative N/A

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

   16. remove-double-neg N/A

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

4. Applied rewrites 74.3%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
6. Step-by-step derivation

7. Applied rewrites 48.4%

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

Alternative 10: 37.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.8 \cdot 10^{+25}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+24}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c -8.8e+25) c (if (<= c 2.05e+24) (* y x) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= -8.8e+25) {
		tmp = c;
	} else if (c <= 2.05e+24) {
		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 (c <= (-8.8d+25)) then
        tmp = c
    else if (c <= 2.05d+24) 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 (c <= -8.8e+25) {
		tmp = c;
	} else if (c <= 2.05e+24) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if c <= -8.8e+25:
		tmp = c
	elif c <= 2.05e+24:
		tmp = y * x
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (c <= -8.8e+25)
		tmp = c;
	elseif (c <= 2.05e+24)
		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 (c <= -8.8e+25)
		tmp = c;
	elseif (c <= 2.05e+24)
		tmp = y * x;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[c, -8.8e+25], c, If[LessEqual[c, 2.05e+24], N[(y * x), $MachinePrecision], c]]

Derivation

1. Split input into 2 regimes

2. if c < -8.8000000000000003e25 or 2.05e24 < c

   1. Initial program 97.3%

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

   2. Taylor expanded in c around inf

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

   4. Applied rewrites 41.0%

      \[\leadsto \color{blue}{c} \]

   if -8.8000000000000003e25 < c < 2.05e24

   1. Initial program 98.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.1

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

   4. Applied rewrites 34.1%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 21.8% accurate, 24.7× 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. Taylor expanded in c around inf

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

4. Applied rewrites 21.8%

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

Reproduce

herbie shell --seed 2025130 
(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))