Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 67.1% → 88.9%

Time: 13.1s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.12e+106)
   (fma (- (- y x)) (/ (- z a) t) y)
   (if (<= t 1.1e+141)
     (fma (/ (- z t) (- a t)) (- y x) x)
     (fma (/ (fma -1.0 y x) t) (- z a) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.12e+106) {
		tmp = fma(-(y - x), ((z - a) / t), y);
	} else if (t <= 1.1e+141) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = fma((fma(-1.0, y, x) / t), (z - a), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.12e+106)
		tmp = fma(Float64(-Float64(y - x)), Float64(Float64(z - a) / t), y);
	elseif (t <= 1.1e+141)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = fma(Float64(fma(-1.0, y, x) / t), Float64(z - a), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.12e+106], N[((-N[(y - x), $MachinePrecision]) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t, 1.1e+141], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-1.0 * y + x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1200000000000001e106

   1. Initial program 34.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 91.2%

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

   if -1.1200000000000001e106 < t < 1.1e141

   1. Initial program 88.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   if 1.1e141 < t

   1. Initial program 37.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 80.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-115}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- z t) y) (- a t)))))
   (if (<= t -1.8e+56)
     (fma (- (- y x)) (/ (- z a) t) y)
     (if (<= t -7e-86)
       t_1
       (if (<= t 3.8e-115)
         (+ x (/ (* (- y x) z) (- a t)))
         (if (<= t 1.82e+137) t_1 (fma (/ (fma -1.0 y x) t) (- z a) y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((z - t) * y) / (a - t));
	double tmp;
	if (t <= -1.8e+56) {
		tmp = fma(-(y - x), ((z - a) / t), y);
	} else if (t <= -7e-86) {
		tmp = t_1;
	} else if (t <= 3.8e-115) {
		tmp = x + (((y - x) * z) / (a - t));
	} else if (t <= 1.82e+137) {
		tmp = t_1;
	} else {
		tmp = fma((fma(-1.0, y, x) / t), (z - a), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t <= -1.8e+56)
		tmp = fma(Float64(-Float64(y - x)), Float64(Float64(z - a) / t), y);
	elseif (t <= -7e-86)
		tmp = t_1;
	elseif (t <= 3.8e-115)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / Float64(a - t)));
	elseif (t <= 1.82e+137)
		tmp = t_1;
	else
		tmp = fma(Float64(fma(-1.0, y, x) / t), Float64(z - a), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8e+56], N[((-N[(y - x), $MachinePrecision]) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t, -7e-86], t$95$1, If[LessEqual[t, 3.8e-115], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.82e+137], t$95$1, N[(N[(N[(-1.0 * y + x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.79999999999999999e56

   1. Initial program 40.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 85.9%

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

   if -1.79999999999999999e56 < t < -7.00000000000000041e-86 or 3.79999999999999992e-115 < t < 1.81999999999999999e137

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -7.00000000000000041e-86 < t < 3.79999999999999992e-115

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if 1.81999999999999999e137 < t

   1. Initial program 41.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 81.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-86}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-115}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+137}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(z - t\right) \cdot y}{a - t}\\ t_2 := \mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-115}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- z t) y) (- a t))))
        (t_2 (fma (- (- y x)) (/ (- z a) t) y)))
   (if (<= t -1.8e+56)
     t_2
     (if (<= t -7e-86)
       t_1
       (if (<= t 3.8e-115)
         (+ x (/ (* (- y x) z) (- a t)))
         (if (<= t 1.82e+137) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((z - t) * y) / (a - t));
	double t_2 = fma(-(y - x), ((z - a) / t), y);
	double tmp;
	if (t <= -1.8e+56) {
		tmp = t_2;
	} else if (t <= -7e-86) {
		tmp = t_1;
	} else if (t <= 3.8e-115) {
		tmp = x + (((y - x) * z) / (a - t));
	} else if (t <= 1.82e+137) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	t_2 = fma(Float64(-Float64(y - x)), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.8e+56)
		tmp = t_2;
	elseif (t <= -7e-86)
		tmp = t_1;
	elseif (t <= 3.8e-115)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / Float64(a - t)));
	elseif (t <= 1.82e+137)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(y - x), $MachinePrecision]) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.8e+56], t$95$2, If[LessEqual[t, -7e-86], t$95$1, If[LessEqual[t, 3.8e-115], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.82e+137], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.79999999999999999e56 or 1.81999999999999999e137 < t

   1. Initial program 40.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 84.3%

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

   if -1.79999999999999999e56 < t < -7.00000000000000041e-86 or 3.79999999999999992e-115 < t < 1.81999999999999999e137

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -7.00000000000000041e-86 < t < 3.79999999999999992e-115

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.7

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

   5. Applied rewrites 89.7%

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

Alternative 4: 52.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.95 \cdot 10^{-162}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z t) (/ y a) x)))
   (if (<= a -4.6e-115)
     t_1
     (if (<= a 3.95e-162)
       (* (/ (- x y) t) z)
       (if (<= a 1.15e-46) (fma 1.0 (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - t), (y / a), x);
	double tmp;
	if (a <= -4.6e-115) {
		tmp = t_1;
	} else if (a <= 3.95e-162) {
		tmp = ((x - y) / t) * z;
	} else if (a <= 1.15e-46) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - t), Float64(y / a), x)
	tmp = 0.0
	if (a <= -4.6e-115)
		tmp = t_1;
	elseif (a <= 3.95e-162)
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	elseif (a <= 1.15e-46)
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -4.6e-115], t$95$1, If[LessEqual[a, 3.95e-162], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 1.15e-46], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.59999999999999969e-115 or 1.15e-46 < a

   1. Initial program 74.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.7%

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

   if -4.59999999999999969e-115 < a < 3.9500000000000001e-162

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 79.7

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 47.9%

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

   if 3.9500000000000001e-162 < a < 1.15e-46

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.95 \cdot 10^{-162}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.55 \cdot 10^{-46}\right):\\ \;\;\;\;x + \frac{z - t}{a} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.25e+66) (not (<= a 1.55e-46)))
   (+ x (* (/ (- z t) a) (- y x)))
   (fma (- (- y x)) (/ (- z a) t) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.25e+66) || !(a <= 1.55e-46)) {
		tmp = x + (((z - t) / a) * (y - x));
	} else {
		tmp = fma(-(y - x), ((z - a) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.25e+66) || !(a <= 1.55e-46))
		tmp = Float64(x + Float64(Float64(Float64(z - t) / a) * Float64(y - x)));
	else
		tmp = fma(Float64(-Float64(y - x)), Float64(Float64(z - a) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.25e+66], N[Not[LessEqual[a, 1.55e-46]], $MachinePrecision]], N[(x + N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(y - x), $MachinePrecision]) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999998e66 or 1.55e-46 < a

   1. Initial program 74.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -1.24999999999999998e66 < a < 1.55e-46

   1. Initial program 71.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 78.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.55 \cdot 10^{-46}\right):\\ \;\;\;\;x + \frac{z - t}{a} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.25e+66) (not (<= a 1.6e-46)))
   (fma (- z t) (/ (- y x) a) x)
   (fma (- (- y x)) (/ (- z a) t) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.25e+66) || !(a <= 1.6e-46)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = fma(-(y - x), ((z - a) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.25e+66) || !(a <= 1.6e-46))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = fma(Float64(-Float64(y - x)), Float64(Float64(z - a) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.25e+66], N[Not[LessEqual[a, 1.6e-46]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[((-N[(y - x), $MachinePrecision]) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999998e66 or 1.6e-46 < a

   1. Initial program 74.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -1.24999999999999998e66 < a < 1.6e-46

   1. Initial program 71.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 78.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(y - x\right), \frac{z - a}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -26000000000000 \lor \neg \left(a \leq 1.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(z - a\right) \cdot \left(y - x\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -26000000000000.0) (not (<= a 1.6e-46)))
   (fma (- z t) (/ (- y x) a) x)
   (- y (/ (* (- z a) (- y x)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -26000000000000.0) || !(a <= 1.6e-46)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = y - (((z - a) * (y - x)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -26000000000000.0) || !(a <= 1.6e-46))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(z - a) * Float64(y - x)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -26000000000000.0], N[Not[LessEqual[a, 1.6e-46]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(z - a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6e13 or 1.6e-46 < a

   1. Initial program 75.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -2.6e13 < a < 1.6e-46

   1. Initial program 70.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 74.7%

      \[\leadsto y - \frac{\left(z - a\right) \cdot \left(y - x\right)}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -26000000000000 \lor \neg \left(a \leq 1.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(z - a\right) \cdot \left(y - x\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.25e+66) (not (<= a 1.6e-46)))
   (fma (- z t) (/ (- y x) a) x)
   (fma (/ (- x y) t) z y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.25e+66) || !(a <= 1.6e-46)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = fma(((x - y) / t), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.25e+66) || !(a <= 1.6e-46))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.25e+66], N[Not[LessEqual[a, 1.6e-46]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999998e66 or 1.6e-46 < a

   1. Initial program 74.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -1.24999999999999998e66 < a < 1.6e-46

   1. Initial program 71.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -1.3e+50)
     t_1
     (if (<= t 2.5e+138)
       (fma (/ y a) z x)
       (if (<= t 3.7e+165) (* x (/ z t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -1.3e+50) {
		tmp = t_1;
	} else if (t <= 2.5e+138) {
		tmp = fma((y / a), z, x);
	} else if (t <= 3.7e+165) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -1.3e+50)
		tmp = t_1;
	elseif (t <= 2.5e+138)
		tmp = fma(Float64(y / a), z, x);
	elseif (t <= 3.7e+165)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.3e+50], t$95$1, If[LessEqual[t, 2.5e+138], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 3.7e+165], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3000000000000001e50 or 3.70000000000000006e165 < t

   1. Initial program 41.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 46.0%

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

   if -1.3000000000000001e50 < t < 2.50000000000000008e138

   1. Initial program 89.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.9%

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

   if 2.50000000000000008e138 < t < 3.70000000000000006e165

   1. Initial program 42.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 70.7%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{x \cdot z}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 70.5%

       \[\leadsto x \cdot \frac{z}{\color{blue}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+113}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -2.3e+32)
     t_1
     (if (<= t 6e+113) (/ (* y z) a) (if (<= t 3.7e+165) (* x (/ z t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -2.3e+32) {
		tmp = t_1;
	} else if (t <= 6e+113) {
		tmp = (y * z) / a;
	} else if (t <= 3.7e+165) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -2.3e+32)
		tmp = t_1;
	elseif (t <= 6e+113)
		tmp = Float64(Float64(y * z) / a);
	elseif (t <= 3.7e+165)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2.3e+32], t$95$1, If[LessEqual[t, 6e+113], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 3.7e+165], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3e32 or 3.70000000000000006e165 < t

   1. Initial program 41.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 45.4%

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

   if -2.3e32 < t < 6e113

   1. Initial program 90.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.0%

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

   if 6e113 < t < 3.70000000000000006e165

   1. Initial program 58.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 51.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{x \cdot z}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.1%

       \[\leadsto x \cdot \frac{z}{\color{blue}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+113}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.55 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.25e+66) (not (<= a 1.55e-46)))
   (fma (/ z a) (- y x) x)
   (fma (/ (- x y) t) z y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.25e+66) || !(a <= 1.55e-46)) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = fma(((x - y) / t), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.25e+66) || !(a <= 1.55e-46))
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.25e+66], N[Not[LessEqual[a, 1.55e-46]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999998e66 or 1.55e-46 < a

   1. Initial program 74.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   if -1.24999999999999998e66 < a < 1.55e-46

   1. Initial program 71.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+66} \lor \neg \left(a \leq 1.55 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.26e+60)
   (fma (- z t) (/ y a) x)
   (if (<= a 1.6e-46) (fma (/ (- x y) t) z y) (fma (/ (- y x) a) z x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.26e+60) {
		tmp = fma((z - t), (y / a), x);
	} else if (a <= 1.6e-46) {
		tmp = fma(((x - y) / t), z, y);
	} else {
		tmp = fma(((y - x) / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.26e+60)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	elseif (a <= 1.6e-46)
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	else
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.26e+60], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.6e-46], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.26e60

   1. Initial program 72.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.0%

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

   if -1.26e60 < a < 1.6e-46

   1. Initial program 71.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.4%

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

   if 1.6e-46 < a

   1. Initial program 76.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.2

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

   5. Applied rewrites 74.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.6e+87)
   (fma 1.0 (- y x) x)
   (if (<= t 1.15e+139) (fma (/ (- y x) a) z x) (* (/ (- z a) t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.6e+87) {
		tmp = fma(1.0, (y - x), x);
	} else if (t <= 1.15e+139) {
		tmp = fma(((y - x) / a), z, x);
	} else {
		tmp = ((z - a) / t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.6e+87)
		tmp = fma(1.0, Float64(y - x), x);
	elseif (t <= 1.15e+139)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	else
		tmp = Float64(Float64(Float64(z - a) / t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.6e+87], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.15e+139], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6e87

   1. Initial program 37.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 52.0%

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

   if -1.6e87 < t < 1.15e139

   1. Initial program 89.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if 1.15e139 < t

   1. Initial program 41.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 40.9%

       \[\leadsto \frac{z - a}{t} \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-115} \lor \neg \left(a \leq 500\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.6e-115) (not (<= a 500.0)))
   (fma (/ y a) z x)
   (* (/ (- x y) t) z)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.6e-115) || !(a <= 500.0)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = ((x - y) / t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.6e-115) || !(a <= 500.0))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.6e-115], N[Not[LessEqual[a, 500.0]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.59999999999999969e-115 or 500 < a

   1. Initial program 73.5%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.8%

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

   if -4.59999999999999969e-115 < a < 500

   1. Initial program 72.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 42.0%

       \[\leadsto \frac{x - y}{t} \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-115} \lor \neg \left(a \leq 500\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.3e+50)
   (fma 1.0 (- y x) x)
   (if (<= t 2.5e+138) (fma (/ y a) z x) (* (/ (- z a) t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.3e+50) {
		tmp = fma(1.0, (y - x), x);
	} else if (t <= 2.5e+138) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = ((z - a) / t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.3e+50)
		tmp = fma(1.0, Float64(y - x), x);
	elseif (t <= 2.5e+138)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(z - a) / t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.3e+50], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.5e+138], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3000000000000001e50

   1. Initial program 42.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 75.5

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 48.2%

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

   if -1.3000000000000001e50 < t < 2.50000000000000008e138

   1. Initial program 89.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.9%

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

   if 2.50000000000000008e138 < t

   1. Initial program 41.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out-- N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 40.9%

       \[\leadsto \frac{z - a}{t} \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+32} \lor \neg \left(t \leq 6.2 \cdot 10^{-36}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.3e+32) (not (<= t 6.2e-36)))
   (fma 1.0 (- y x) x)
   (/ (* y z) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.3e+32) || !(t <= 6.2e-36)) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = (y * z) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.3e+32) || !(t <= 6.2e-36))
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = Float64(Float64(y * z) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.3e+32], N[Not[LessEqual[t, 6.2e-36]], $MachinePrecision]], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.3e32 or 6.1999999999999997e-36 < t

   1. Initial program 51.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 36.1%

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

   if -2.3e32 < t < 6.1999999999999997e-36

   1. Initial program 92.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.2%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+32} \lor \neg \left(t \leq 6.2 \cdot 10^{-36}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 19.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, y - x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 86.5

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

4. Applied rewrites 86.5%

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

5. Taylor expanded in t around inf

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

7. Applied rewrites 20.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
8. Add Preprocessing

Alternative 18: 2.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, -x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 86.5

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

4. Applied rewrites 86.5%

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

5. Taylor expanded in t around inf

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

7. Applied rewrites 20.7%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(1, \color{blue}{-1 \cdot x}, x\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 2.8

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

10. Applied rewrites 2.8%

    \[\leadsto \mathsf{fma}\left(1, \color{blue}{-x}, x\right) \]
11. Add Preprocessing

Developer Target 1: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025008 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))