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

Percentage Accurate: 75.9% → 93.3%

Time: 5.0s

Alternatives: 12

Speedup: 0.9×

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: 75.9% 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: 93.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + (y * ((1.0 + (t / (a - t))) - (z / (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 * ((1.0d0 + (t / (a - t))) - (z / (a - t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

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 93.3

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

4. Applied rewrites 93.3%

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

Alternative 2: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ t_3 := \left(x + y\right) - \frac{y \cdot z}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-212}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+297}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-1}{\frac{a - t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (* -1.0 (/ (- a z) t)) y x))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t))))
        (t_3 (- (+ x y) (/ (* y z) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -4e-212)
       t_3
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 1e+297) t_3 (+ x (* y (/ -1.0 (/ (- a t) z))))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-1.0 * ((a - z) / t)), y, x);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double t_3 = (x + y) - ((y * z) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -4e-212) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+297) {
		tmp = t_3;
	} else {
		tmp = x + (y * (-1.0 / ((a - t) / z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	t_3 = Float64(Float64(x + y) - Float64(Float64(y * z) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -4e-212)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+297)
		tmp = t_3;
	else
		tmp = Float64(x + Float64(y * Float64(-1.0 / Float64(Float64(a - t) / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $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]}, Block[{t$95$3 = N[(N[(x + y), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -4e-212], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 1e+297], t$95$3, N[(x + N[(y * N[(-1.0 / N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or -3.99999999999999982e-212 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 61.1

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

   12. Applied rewrites 61.1%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -3.99999999999999982e-212 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1e297

   1. Initial program 75.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if 1e297 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 76.7

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

   9. Applied rewrites 76.7%

      \[\leadsto x + y \cdot \frac{-1}{\frac{a - t}{\color{blue}{z}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ t_3 := \left(x + y\right) - \frac{y \cdot z}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-212}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+297}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (* -1.0 (/ (- a z) t)) y x))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t))))
        (t_3 (- (+ x y) (/ (* y z) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -4e-212)
       t_3
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 1e+297) t_3 (fma (/ z (- t a)) y x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-1.0 * ((a - z) / t)), y, x);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double t_3 = (x + y) - ((y * z) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -4e-212) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+297) {
		tmp = t_3;
	} else {
		tmp = fma((z / (t - a)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	t_3 = Float64(Float64(x + y) - Float64(Float64(y * z) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -4e-212)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+297)
		tmp = t_3;
	else
		tmp = fma(Float64(z / Float64(t - a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $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]}, Block[{t$95$3 = N[(N[(x + y), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -4e-212], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 1e+297], t$95$3, N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or -3.99999999999999982e-212 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 61.1

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

   12. Applied rewrites 61.1%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -3.99999999999999982e-212 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1e297

   1. Initial program 75.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if 1e297 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

Alternative 4: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a - t}, y + x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+297}:\\ \;\;\;\;\left(x + y\right) - \frac{y \cdot z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-1}{\frac{a - t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_1 -4e-212)
     (fma (- t z) (/ y (- a t)) (+ y x))
     (if (<= t_1 0.0)
       (fma (* -1.0 (/ (- a z) t)) y x)
       (if (<= t_1 1e+297)
         (- (+ x y) (/ (* y z) (- a t)))
         (+ x (* y (/ -1.0 (/ (- a 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 <= -4e-212) {
		tmp = fma((t - z), (y / (a - t)), (y + x));
	} else if (t_1 <= 0.0) {
		tmp = fma((-1.0 * ((a - z) / t)), y, x);
	} else if (t_1 <= 1e+297) {
		tmp = (x + y) - ((y * z) / (a - t));
	} else {
		tmp = x + (y * (-1.0 / ((a - 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 <= -4e-212)
		tmp = fma(Float64(t - z), Float64(y / Float64(a - t)), Float64(y + x));
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x);
	elseif (t_1 <= 1e+297)
		tmp = Float64(Float64(x + y) - Float64(Float64(y * z) / Float64(a - t)));
	else
		tmp = Float64(x + Float64(y * Float64(-1.0 / Float64(Float64(a - t) / z))));
	end
	return 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, -4e-212], N[(N[(t - z), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+297], N[(N[(x + y), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(-1.0 / N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -3.99999999999999982e-212

   1. Initial program 75.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 82.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 82.4

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

   3. Applied rewrites 82.4%

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

   if -3.99999999999999982e-212 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 61.1

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

   12. Applied rewrites 61.1%

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1e297

   1. Initial program 75.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if 1e297 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 76.7

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

   9. Applied rewrites 76.7%

      \[\leadsto x + y \cdot \frac{-1}{\frac{a - t}{\color{blue}{z}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, z - t, x\right) + y\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+297}:\\ \;\;\;\;\left(x + y\right) - \frac{y \cdot z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-1}{\frac{a - t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_1 -4e-212)
     (+ (fma (/ y (- t a)) (- z t) x) y)
     (if (<= t_1 0.0)
       (fma (* -1.0 (/ (- a z) t)) y x)
       (if (<= t_1 1e+297)
         (- (+ x y) (/ (* y z) (- a t)))
         (+ x (* y (/ -1.0 (/ (- a 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 <= -4e-212) {
		tmp = fma((y / (t - a)), (z - t), x) + y;
	} else if (t_1 <= 0.0) {
		tmp = fma((-1.0 * ((a - z) / t)), y, x);
	} else if (t_1 <= 1e+297) {
		tmp = (x + y) - ((y * z) / (a - t));
	} else {
		tmp = x + (y * (-1.0 / ((a - 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 <= -4e-212)
		tmp = Float64(fma(Float64(y / Float64(t - a)), Float64(z - t), x) + y);
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x);
	elseif (t_1 <= 1e+297)
		tmp = Float64(Float64(x + y) - Float64(Float64(y * z) / Float64(a - t)));
	else
		tmp = Float64(x + Float64(y * Float64(-1.0 / Float64(Float64(a - t) / z))));
	end
	return 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, -4e-212], N[(N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+297], N[(N[(x + y), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(-1.0 / N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -3.99999999999999982e-212

   1. Initial program 75.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 82.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 82.4

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

   3. Applied rewrites 82.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

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

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

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

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

      12. lower-fma.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   if -3.99999999999999982e-212 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 61.1

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

   12. Applied rewrites 61.1%

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1e297

   1. Initial program 75.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if 1e297 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 76.7

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

   9. Applied rewrites 76.7%

      \[\leadsto x + y \cdot \frac{-1}{\frac{a - t}{\color{blue}{z}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - \frac{y}{a - t} \cdot \left(z - t\right)\right) + x\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y (* (/ y (- a t)) (- z t))) x))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 -4e-212)
     t_1
     (if (<= t_2 0.0) (fma (* -1.0 (/ (- a z) t)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - ((y / (a - t)) * (z - t))) + x;
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -4e-212) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((-1.0 * ((a - z) / t)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(Float64(y / Float64(a - t)) * Float64(z - t))) + x)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -4e-212)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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, -4e-212], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y + x), $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))) < -3.99999999999999982e-212 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 78.8

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.7

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

   3. Applied rewrites 86.7%

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

   if -3.99999999999999982e-212 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 61.1

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

   12. Applied rewrites 61.1%

       \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - \frac{z}{a} \cdot y\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (* (/ z a) y))))
   (if (<= a -1.05e+65) t_1 (if (<= a 3.6e-36) (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 / a) * y);
	double tmp;
	if (a <= -1.05e+65) {
		tmp = t_1;
	} else if (a <= 3.6e-36) {
		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(Float64(z / a) * y))
	tmp = 0.0
	if (a <= -1.05e+65)
		tmp = t_1;
	elseif (a <= 3.6e-36)
		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[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.05e+65], t$95$1, If[LessEqual[a, 3.6e-36], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.04999999999999996e65 or 3.60000000000000032e-36 < a

   1. Initial program 75.9%

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

   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 63.6

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

   4. Applied rewrites 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.7

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

   6. Applied rewrites 65.7%

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

   if -1.04999999999999996e65 < a < 3.60000000000000032e-36

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

Alternative 8: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.2e+74)
   (fma 1.0 y x)
   (if (<= a 1.4e-33) (fma (/ z (- t a)) y x) (- (+ x y) (* z (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.2e+74) {
		tmp = fma(1.0, y, x);
	} else if (a <= 1.4e-33) {
		tmp = fma((z / (t - a)), y, x);
	} else {
		tmp = (x + y) - (z * (y / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.2e+74)
		tmp = fma(1.0, y, x);
	elseif (a <= 1.4e-33)
		tmp = fma(Float64(z / Float64(t - a)), y, x);
	else
		tmp = Float64(Float64(x + y) - Float64(z * Float64(y / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.2e+74], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[a, 1.4e-33], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.1999999999999998e74

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if -4.1999999999999998e74 < a < 1.4e-33

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   if 1.4e-33 < a

   1. Initial program 75.9%

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

   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 63.6

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

   4. Applied rewrites 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.7

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

   6. Applied rewrites 65.7%

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

Alternative 9: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;a \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.2e+74)
   (fma 1.0 y x)
   (if (<= a 1e+64) (fma (/ z (- t a)) y x) (fma 1.0 y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.2e+74) {
		tmp = fma(1.0, y, x);
	} else if (a <= 1e+64) {
		tmp = fma((z / (t - a)), y, x);
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.2e+74)
		tmp = fma(1.0, y, x);
	elseif (a <= 1e+64)
		tmp = fma(Float64(z / Float64(t - a)), y, x);
	else
		tmp = fma(1.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.2e+74], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[a, 1e+64], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1999999999999998e74 or 1.00000000000000002e64 < a

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if -4.1999999999999998e74 < a < 1.00000000000000002e64

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

Alternative 10: 75.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -8.6e+48)
   (fma 1.0 y x)
   (if (<= a 1.4e-33) (fma (/ z t) y x) (fma 1.0 y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -8.6e+48) {
		tmp = fma(1.0, y, x);
	} else if (a <= 1.4e-33) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -8.6e+48)
		tmp = fma(1.0, y, x);
	elseif (a <= 1.4e-33)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = fma(1.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -8.6e+48], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[a, 1.4e-33], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.59999999999999957e48 or 1.4e-33 < a

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if -8.59999999999999957e48 < a < 1.4e-33

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 62.3

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

   7. Applied rewrites 62.3%

      \[\leadsto x + y \cdot \frac{z}{\color{blue}{t}} \]
   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.3

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

   9. Applied rewrites 62.3%

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

Alternative 11: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_1 -4e-212)
     (fma 1.0 y x)
     (if (<= t_1 0.0) (* 1.0 x) (fma 1.0 y x)))))

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 <= -4e-212) {
		tmp = fma(1.0, y, x);
	} else if (t_1 <= 0.0) {
		tmp = 1.0 * x;
	} else {
		tmp = fma(1.0, y, x);
	}
	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 <= -4e-212)
		tmp = fma(1.0, y, x);
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 * x);
	else
		tmp = fma(1.0, y, x);
	end
	return 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, -4e-212], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 * x), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -3.99999999999999982e-212 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 75.9%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.7

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if -3.99999999999999982e-212 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sum-to-mult N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 73.1

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 79.0

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

   3. Applied rewrites 79.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a):
	return 1.0 * x

Julia

function code(x, y, z, t, a)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. sum-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 73.1

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 79.0

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 50.3%

   \[\leadsto \color{blue}{1} \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025148 
(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))))