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

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. sub-flip-reverse N/A

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

   6. lower--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mult-flip N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 100.0

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

3. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--r+ N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sign-sub N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. mult-flip-rev N/A

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

   12. associate-/l* N/A

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

   13. lift-*.f64 N/A

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

   14. frac-2neg N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

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

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. +-commutative N/A

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

   23. lift-*.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 100.0

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

3. Applied rewrites 100.0%

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

Alternative 3: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -0.5, y, x \cdot 0.125\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (<= t_1 -4e+31)
     (fma (* z -0.5) y (* x 0.125))
     (if (<= t_1 5e+45) (fma x 0.125 t) (fma (* -0.5 z) y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if (t_1 <= -4e+31) {
		tmp = fma((z * -0.5), y, (x * 0.125));
	} else if (t_1 <= 5e+45) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = fma((-0.5 * z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if (t_1 <= -4e+31)
		tmp = fma(Float64(z * -0.5), y, Float64(x * 0.125));
	elseif (t_1 <= 5e+45)
		tmp = fma(x, 0.125, t);
	else
		tmp = fma(Float64(-0.5 * z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+31], N[(N[(z * -0.5), $MachinePrecision] * y + N[(x * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+45], N[(x * 0.125 + t), $MachinePrecision], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -3.9999999999999999e31

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   3. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. mult-flip-rev N/A

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

      12. associate-/l* N/A

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

      13. lift-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

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

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 68.3

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

   8. Applied rewrites 68.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.3

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

   10. Applied rewrites 68.3%

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

   if -3.9999999999999999e31 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 5e45

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.6

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

   4. Applied rewrites 64.6%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 64.6

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

   6. Applied rewrites 64.6%

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

   if 5e45 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 67.6

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

   4. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 67.6

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

   6. Applied rewrites 67.6%

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

Alternative 4: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (<= t_1 -4e+31)
     (fma -0.5 (* y z) (* 0.125 x))
     (if (<= t_1 5e+45) (fma x 0.125 t) (fma (* -0.5 z) y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if (t_1 <= -4e+31) {
		tmp = fma(-0.5, (y * z), (0.125 * x));
	} else if (t_1 <= 5e+45) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = fma((-0.5 * z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if (t_1 <= -4e+31)
		tmp = fma(-0.5, Float64(y * z), Float64(0.125 * x));
	elseif (t_1 <= 5e+45)
		tmp = fma(x, 0.125, t);
	else
		tmp = fma(Float64(-0.5 * z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+31], N[(-0.5 * N[(y * z), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+45], N[(x * 0.125 + t), $MachinePrecision], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -3.9999999999999999e31

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 68.3

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

   6. Applied rewrites 68.3%

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

   if -3.9999999999999999e31 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 5e45

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.6

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

   4. Applied rewrites 64.6%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 64.6

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

   6. Applied rewrites 64.6%

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

   if 5e45 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 67.6

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

   4. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 67.6

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

   6. Applied rewrites 67.6%

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

Alternative 5: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ t_2 := \mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)) (t_2 (fma (* -0.5 z) y t)))
   (if (<= t_1 -4e+31) t_2 (if (<= t_1 5e+45) (fma x 0.125 t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double t_2 = fma((-0.5 * z), y, t);
	double tmp;
	if (t_1 <= -4e+31) {
		tmp = t_2;
	} else if (t_1 <= 5e+45) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	t_2 = fma(Float64(-0.5 * z), y, t)
	tmp = 0.0
	if (t_1 <= -4e+31)
		tmp = t_2;
	elseif (t_1 <= 5e+45)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+31], t$95$2, If[LessEqual[t$95$1, 5e+45], N[(x * 0.125 + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -3.9999999999999999e31 or 5e45 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 67.6

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

   4. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 67.6

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

   6. Applied rewrites 67.6%

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

   if -3.9999999999999999e31 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 5e45

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.6

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

   4. Applied rewrites 64.6%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 64.6

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

   6. Applied rewrites 64.6%

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

Alternative 6: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ t_2 := -0.5 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+118}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)) (t_2 (* -0.5 (* y z))))
   (if (<= t_1 -5e+118) t_2 (if (<= t_1 1e+177) (fma x 0.125 t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double t_2 = -0.5 * (y * z);
	double tmp;
	if (t_1 <= -5e+118) {
		tmp = t_2;
	} else if (t_1 <= 1e+177) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	t_2 = Float64(-0.5 * Float64(y * z))
	tmp = 0.0
	if (t_1 <= -5e+118)
		tmp = t_2;
	elseif (t_1 <= 1e+177)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+118], t$95$2, If[LessEqual[t$95$1, 1e+177], N[(x * 0.125 + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -4.99999999999999972e118 or 1e177 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 36.8

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

   6. Applied rewrites 36.8%

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

   if -4.99999999999999972e118 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 1e177

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.6

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

   4. Applied rewrites 64.6%

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 64.6

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

   6. Applied rewrites 64.6%

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

Alternative 7: 64.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 64.6

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

4. Applied rewrites 64.6%

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

   1. metadata-eval N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 64.6

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

6. Applied rewrites 64.6%

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

Alternative 8: 49.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+67}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+106}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.2e+67) t (if (<= t 4.6e+106) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.2e+67) {
		tmp = t;
	} else if (t <= 4.6e+106) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.2d+67)) then
        tmp = t
    else if (t <= 4.6d+106) then
        tmp = 0.125d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.2e+67) {
		tmp = t;
	} else if (t <= 4.6e+106) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.2e+67:
		tmp = t
	elif t <= 4.6e+106:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.2e+67)
		tmp = t;
	elseif (t <= 4.6e+106)
		tmp = Float64(0.125 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.2e+67)
		tmp = t;
	elseif (t <= 4.6e+106)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.2e+67], t, If[LessEqual[t, 4.6e+106], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2000000000000003e67 or 4.6000000000000004e106 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.6

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 32.8%

      \[\leadsto t \]

   if -4.2000000000000003e67 < t < 4.6000000000000004e106

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   3. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. mult-flip-rev N/A

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

      12. associate-/l* N/A

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

      13. lift-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

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

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval 33.5

         \[\leadsto 0.125 \cdot x \]

   8. Applied rewrites 33.5%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 32.8% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 64.6

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

4. Applied rewrites 64.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 32.8%

   \[\leadsto t \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64
  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))