Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Percentage Accurate: 76.5% → 98.5%

Time: 4.8s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (+ x y) (/ (* (- z t) y) (- a t))) 2e+32)
   (fma (/ (- z a) (- t a)) y x)
   (fma (/ y (- t a)) (- z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= 2e+32) {
		tmp = fma(((z - a) / (t - a)), y, x);
	} else {
		tmp = fma((y / (t - a)), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t))) <= 2e+32)
		tmp = fma(Float64(Float64(z - a) / Float64(t - a)), y, x);
	else
		tmp = fma(Float64(y / Float64(t - a)), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+32], N[(N[(N[(z - a), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 2.00000000000000011e32

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.3

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

   8. Applied rewrites 98.3%

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

   if 2.00000000000000011e32 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

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

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

   8. Applied rewrites 95.9%

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

Alternative 2: 95.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ y (- t a)) (- z a) x))

C

double code(double x, double y, double z, double t, double a) {
	return fma((y / (t - a)), (z - a), x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(y / Float64(t - a)), Float64(z - a), x)
end

Wolfram

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

Derivation

1. Initial program 76.5%

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

2. Taylor expanded in y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 93.4

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

4. Applied rewrites 93.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. add-to-fraction N/A

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

   5. mult-flip N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. distribute-rgt-out-- N/A

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

   9. *-lft-identity N/A

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

   10. sum-to-mult-rev N/A

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

   11. lift-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. frac-2neg N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

6. Applied rewrites 93.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift--.f64 N/A

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

   7. sub-negate-rev N/A

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

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

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

   9. sub-negate-rev N/A

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

   10. lift--.f64 N/A

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

   11. mul-1-neg N/A

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

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

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

8. Applied rewrites 95.9%

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

Alternative 3: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-227}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- t a)) y x))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 -2e+306)
     t_1
     (if (<= t_2 -5e-227)
       (+ x y)
       (if (<= t_2 0.0) (fma (/ (- z a) t) y x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / (t - a)), y, x);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= -5e-227) {
		tmp = x + y;
	} else if (t_2 <= 0.0) {
		tmp = fma(((z - a) / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(t - a)), y, x)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= -5e-227)
		tmp = Float64(x + y);
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+306], t$95$1, If[LessEqual[t$95$2, -5e-227], N[(x + y), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -2.00000000000000003e306 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x + y \cdot \frac{z}{t - \color{blue}{a}} \]

      2. lower--.f64 76.2

         \[\leadsto x + y \cdot \frac{z}{t - a} \]

   9. Applied rewrites 76.2%

      \[\leadsto x + y \cdot \frac{z}{\color{blue}{t - a}} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + \color{blue}{y \cdot \frac{z}{t - a}} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \frac{z}{t - a} + \color{blue}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{z}{t - a} + x \]

       4. *-commutative N/A

          \[\leadsto \frac{z}{t - a} \cdot y + x \]

       5. lower-fma.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -2.00000000000000003e306 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -4.99999999999999961e-227

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-+.f64 60.2

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

   11. Applied rewrites 60.2%

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

   if -4.99999999999999961e-227 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 76.5%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{a \cdot y - y \cdot z}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      6. lower-*.f64 57.7

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{a \cdot y - y \cdot z}{t} + \color{blue}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{a \cdot y - y \cdot z}{t} + x \]

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. sub-negate N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 61.1

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

   6. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 60.7%

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

Alternative 4: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - z \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (* z (/ y a)))))
   (if (<= a -3.8e+66) t_1 (if (<= a 9.5e-8) (fma (/ z (- t a)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - (z * (y / a));
	double tmp;
	if (a <= -3.8e+66) {
		tmp = t_1;
	} else if (a <= 9.5e-8) {
		tmp = fma((z / (t - a)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x + y) - Float64(z * Float64(y / a)))
	tmp = 0.0
	if (a <= -3.8e+66)
		tmp = t_1;
	elseif (a <= 9.5e-8)
		tmp = fma(Float64(z / Float64(t - a)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.8e+66], t$95$1, If[LessEqual[a, 9.5e-8], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8000000000000002e66 or 9.50000000000000036e-8 < a

   1. Initial program 76.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 64.8

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

   4. Applied rewrites 64.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 66.4

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

   6. Applied rewrites 66.4%

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

   if -3.8000000000000002e66 < a < 9.50000000000000036e-8

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x + y \cdot \frac{z}{t - \color{blue}{a}} \]

      2. lower--.f64 76.2

         \[\leadsto x + y \cdot \frac{z}{t - a} \]

   9. Applied rewrites 76.2%

      \[\leadsto x + y \cdot \frac{z}{\color{blue}{t - a}} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + \color{blue}{y \cdot \frac{z}{t - a}} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \frac{z}{t - a} + \color{blue}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{z}{t - a} + x \]

       4. *-commutative N/A

          \[\leadsto \frac{z}{t - a} \cdot y + x \]

       5. lower-fma.f64 76.2

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

   11. Applied rewrites 76.2%

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

Alternative 5: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+70}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.6e+70)
   (+ x y)
   (if (<= a 1.65e-7) (fma (/ z (- t a)) y x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.6e+70) {
		tmp = x + y;
	} else if (a <= 1.65e-7) {
		tmp = fma((z / (t - a)), y, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.6e+70)
		tmp = Float64(x + y);
	elseif (a <= 1.65e-7)
		tmp = fma(Float64(z / Float64(t - a)), y, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.6e+70], N[(x + y), $MachinePrecision], If[LessEqual[a, 1.65e-7], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6e70 or 1.6500000000000001e-7 < a

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-+.f64 60.2

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

   11. Applied rewrites 60.2%

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

   if -2.6e70 < a < 1.6500000000000001e-7

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x + y \cdot \frac{z}{t - \color{blue}{a}} \]

      2. lower--.f64 76.2

         \[\leadsto x + y \cdot \frac{z}{t - a} \]

   9. Applied rewrites 76.2%

      \[\leadsto x + y \cdot \frac{z}{\color{blue}{t - a}} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + \color{blue}{y \cdot \frac{z}{t - a}} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \frac{z}{t - a} + \color{blue}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{z}{t - a} + x \]

       4. *-commutative N/A

          \[\leadsto \frac{z}{t - a} \cdot y + x \]

       5. lower-fma.f64 76.2

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

   11. Applied rewrites 76.2%

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

Alternative 6: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2100:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2100.0) (+ x y) (if (<= a 2.4e-8) (fma (/ z t) y x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2100.0) {
		tmp = x + y;
	} else if (a <= 2.4e-8) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2100.0)
		tmp = Float64(x + y);
	elseif (a <= 2.4e-8)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2100.0], N[(x + y), $MachinePrecision], If[LessEqual[a, 2.4e-8], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2100 or 2.39999999999999998e-8 < a

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-+.f64 60.2

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

   11. Applied rewrites 60.2%

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

   if -2100 < a < 2.39999999999999998e-8

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in a around 0

      \[\leadsto x + y \cdot \frac{z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 61.7

         \[\leadsto x + y \cdot \frac{z}{t} \]

   7. Applied rewrites 61.7%

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

      1. lift-+.f64 N/A

         \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \frac{z}{t} + \color{blue}{x} \]

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

         \[\leadsto \frac{z}{t} \cdot y + x \]

      8. lower-fma.f64 61.7

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

   9. Applied rewrites 61.7%

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

Alternative 7: 64.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_1 -2e+306)
     (* (/ z t) y)
     (if (<= t_1 2e+305) (+ x y) (* (/ y t) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = (z / t) * y;
	} else if (t_1 <= 2e+305) {
		tmp = x + y;
	} else {
		tmp = (y / t) * z;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x + y) - (((z - t) * y) / (a - t))
    if (t_1 <= (-2d+306)) then
        tmp = (z / t) * y
    else if (t_1 <= 2d+305) then
        tmp = x + y
    else
        tmp = (y / t) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = (z / t) * y;
	} else if (t_1 <= 2e+305) {
		tmp = x + y;
	} else {
		tmp = (y / t) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_1 <= -2e+306:
		tmp = (z / t) * y
	elif t_1 <= 2e+305:
		tmp = x + y
	else:
		tmp = (y / t) * z
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(Float64(z / t) * y);
	elseif (t_1 <= 2e+305)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(y / t) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_1 <= -2e+306)
		tmp = (z / t) * y;
	elseif (t_1 <= 2e+305)
		tmp = x + y;
	else
		tmp = (y / t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+306], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[(x + y), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -2.00000000000000003e306

   1. Initial program 76.5%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{a \cdot y - y \cdot z}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      6. lower-*.f64 57.7

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      2. lower-*.f64 18.6

         \[\leadsto \frac{y \cdot z}{t} \]

   7. Applied rewrites 18.6%

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{t}} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{t} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z}{t} \cdot y \]

      6. lower-/.f64 20.2

         \[\leadsto \frac{z}{t} \cdot y \]

   9. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]

   if -2.00000000000000003e306 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1.9999999999999999e305

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-+.f64 60.2

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

   11. Applied rewrites 60.2%

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

   if 1.9999999999999999e305 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 76.5%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{a \cdot y - y \cdot z}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      6. lower-*.f64 57.7

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      2. lower-*.f64 18.6

         \[\leadsto \frac{y \cdot z}{t} \]

   7. Applied rewrites 18.6%

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y}{t} \cdot z \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{y}{t} \cdot z \]

      5. lower-*.f64 19.9

         \[\leadsto \frac{y}{t} \cdot z \]

   9. Applied rewrites 19.9%

      \[\leadsto \frac{y}{t} \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y t) z)) (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 -2e+306) t_1 (if (<= t_2 2e+305) (+ x y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / t) * z;
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= 2e+305) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y / t) * z
    t_2 = (x + y) - (((z - t) * y) / (a - t))
    if (t_2 <= (-2d+306)) then
        tmp = t_1
    else if (t_2 <= 2d+305) then
        tmp = x + y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / t) * z;
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= 2e+305) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / t) * z
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 <= -2e+306:
		tmp = t_1
	elif t_2 <= 2e+305:
		tmp = x + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / t) * z)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= 2e+305)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / t) * z;
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= 2e+305)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+306], t$95$1, If[LessEqual[t$95$2, 2e+305], N[(x + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -2.00000000000000003e306 or 1.9999999999999999e305 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 76.5%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{a \cdot y - y \cdot z}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

      6. lower-*.f64 57.7

         \[\leadsto x + -1 \cdot \frac{a \cdot y - y \cdot z}{t} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{a \cdot y - y \cdot z}{t}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      2. lower-*.f64 18.6

         \[\leadsto \frac{y \cdot z}{t} \]

   7. Applied rewrites 18.6%

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{t} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y}{t} \cdot z \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{y}{t} \cdot z \]

      5. lower-*.f64 19.9

         \[\leadsto \frac{y}{t} \cdot z \]

   9. Applied rewrites 19.9%

      \[\leadsto \frac{y}{t} \cdot z \]

   if -2.00000000000000003e306 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1.9999999999999999e305

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-lft-identity N/A

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

      10. sum-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-+.f64 60.2

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

   11. Applied rewrites 60.2%

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

Alternative 9: 60.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
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
    code = x + y
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return Float64(x + y)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

2. Taylor expanded in y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 93.4

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

4. Applied rewrites 93.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. add-to-fraction N/A

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

   5. mult-flip N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. distribute-rgt-out-- N/A

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

   9. *-lft-identity N/A

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

   10. sum-to-mult-rev N/A

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

   11. lift-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. frac-2neg N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

6. Applied rewrites 93.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 93.3

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

8. Applied rewrites 98.3%

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

9. Taylor expanded in a around inf

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

    1. lower-+.f64 60.2

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

11. Applied rewrites 60.2%

    \[\leadsto \color{blue}{x + y} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64
  (- (+ x y) (/ (* (- z t) y) (- a t))))