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

Percentage Accurate: 67.0% → 85.8%

Time: 9.7s

Alternatives: 11

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: 67.0% 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: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.8 \cdot 10^{-126} \lor \neg \left(a \leq 4.4 \cdot 10^{-225}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -8.8e-126) (not (<= a 4.4e-225)))
   (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 ((a <= -8.8e-126) || !(a <= 4.4e-225)) {
		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 ((a <= -8.8e-126) || !(a <= 4.4e-225))
		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[Or[LessEqual[a, -8.8e-126], N[Not[LessEqual[a, 4.4e-225]], $MachinePrecision]], 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 2 regimes

2. if a < -8.80000000000000058e-126 or 4.4e-225 < a

   1. Initial program 74.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 91.0

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

   4. Applied rewrites 91.0%

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

   if -8.80000000000000058e-126 < a < 4.4e-225

   1. Initial program 57.4%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.8 \cdot 10^{-126} \lor \neg \left(a \leq 4.4 \cdot 10^{-225}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - z}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) (- a t)) y x)))
   (if (<= y -1.45e-47)
     t_1
     (if (<= y -2.6e-144)
       (fma (/ x t) (- z a) y)
       (if (<= y 2.1e-26) (fma x (/ (- t z) (- a t)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / (a - t)), y, x);
	double tmp;
	if (y <= -1.45e-47) {
		tmp = t_1;
	} else if (y <= -2.6e-144) {
		tmp = fma((x / t), (z - a), y);
	} else if (y <= 2.1e-26) {
		tmp = fma(x, ((t - z) / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / Float64(a - t)), y, x)
	tmp = 0.0
	if (y <= -1.45e-47)
		tmp = t_1;
	elseif (y <= -2.6e-144)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	elseif (y <= 2.1e-26)
		tmp = fma(x, Float64(Float64(t - z) / Float64(a - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[y, -1.45e-47], t$95$1, If[LessEqual[y, -2.6e-144], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 2.1e-26], N[(x * N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e-47 or 2.10000000000000008e-26 < y

   1. Initial program 66.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 80.1%

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

   if -1.45e-47 < y < -2.6000000000000001e-144

   1. Initial program 78.1%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.4%

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

   if -2.6000000000000001e-144 < y < 2.10000000000000008e-26

   1. Initial program 78.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 75.0

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

   10. Applied rewrites 75.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t - z}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-96}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x t) (- z a) y)))
   (if (<= t -2.8e+91)
     t_1
     (if (<= t -2.1e-96)
       (* (- y x) (/ z (- a t)))
       (if (<= t 1.15e+55) (fma (/ z a) (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / t), (z - a), y);
	double tmp;
	if (t <= -2.8e+91) {
		tmp = t_1;
	} else if (t <= -2.1e-96) {
		tmp = (y - x) * (z / (a - t));
	} else if (t <= 1.15e+55) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -2.8e+91)
		tmp = t_1;
	elseif (t <= -2.1e-96)
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	elseif (t <= 1.15e+55)
		tmp = fma(Float64(z / a), 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[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.8e+91], t$95$1, If[LessEqual[t, -2.1e-96], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+55], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7999999999999999e91 or 1.14999999999999994e55 < t

   1. Initial program 42.2%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.3%

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

   if -2.7999999999999999e91 < t < -2.10000000000000001e-96

   1. Initial program 77.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -2.10000000000000001e-96 < t < 1.14999999999999994e55

   1. Initial program 89.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 94.4

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

   4. Applied rewrites 94.4%

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

   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 79.5

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

   7. Applied rewrites 79.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-96}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-33}:\\ \;\;\;\;x + \frac{z - t}{a} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.5e-33)
   (+ x (* (/ (- z t) a) (- y x)))
   (if (<= a 9.8e-222)
     (fma (/ (fma -1.0 y x) t) (- z a) y)
     (fma (/ (- z t) (- a t)) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.5e-33) {
		tmp = x + (((z - t) / a) * (y - x));
	} else if (a <= 9.8e-222) {
		tmp = fma((fma(-1.0, y, x) / t), (z - a), y);
	} else {
		tmp = fma(((z - t) / (a - t)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.5e-33)
		tmp = Float64(x + Float64(Float64(Float64(z - t) / a) * Float64(y - x)));
	elseif (a <= 9.8e-222)
		tmp = fma(Float64(fma(-1.0, y, x) / t), Float64(z - a), y);
	else
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.5e-33], N[(x + N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.8e-222], N[(N[(N[(-1.0 * y + x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.5e-33

   1. Initial program 71.4%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -5.5e-33 < a < 9.7999999999999999e-222

   1. Initial program 63.9%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 84.2%

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

   if 9.7999999999999999e-222 < a

   1. Initial program 76.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 79.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-33}:\\ \;\;\;\;x + \frac{z - t}{a} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, y, x\right)}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+63} \lor \neg \left(t \leq 1.15 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.6e+63) (not (<= t 1.15e+55)))
   (fma (/ x t) (- z a) y)
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.6e+63) || !(t <= 1.15e+55)) {
		tmp = fma((x / t), (z - a), y);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.6e+63) || !(t <= 1.15e+55))
		tmp = fma(Float64(x / t), Float64(z - a), y);
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.6e+63], N[Not[LessEqual[t, 1.15e+55]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.60000000000000006e63 or 1.14999999999999994e55 < t

   1. Initial program 42.2%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.5%

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

   if -1.60000000000000006e63 < t < 1.14999999999999994e55

   1. Initial program 88.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 94.2

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

   4. Applied rewrites 94.2%

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

   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 74.2

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

   7. Applied rewrites 74.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+63} \lor \neg \left(t \leq 1.15 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+63} \lor \neg \left(t \leq 1.15 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.6e+63) (not (<= t 1.15e+55)))
   (fma (/ x t) (- z a) y)
   (fma (/ (- y x) a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.6e+63) || !(t <= 1.15e+55)) {
		tmp = fma((x / t), (z - a), y);
	} else {
		tmp = fma(((y - x) / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.6e+63) || !(t <= 1.15e+55))
		tmp = fma(Float64(x / t), Float64(z - a), y);
	else
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.6e+63], N[Not[LessEqual[t, 1.15e+55]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.60000000000000006e63 or 1.14999999999999994e55 < t

   1. Initial program 42.2%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.5%

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

   if -1.60000000000000006e63 < t < 1.14999999999999994e55

   1. Initial program 88.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 72.5

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

   5. Applied rewrites 72.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+63} \lor \neg \left(t \leq 1.15 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-53} \lor \neg \left(t \leq 1.1 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.1e-53) (not (<= t 1.1e+55)))
   (fma (/ x t) (- z a) y)
   (fma (/ z a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.1e-53) || !(t <= 1.1e+55)) {
		tmp = fma((x / t), (z - a), y);
	} else {
		tmp = fma((z / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.1e-53) || !(t <= 1.1e+55))
		tmp = fma(Float64(x / t), Float64(z - a), y);
	else
		tmp = fma(Float64(z / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.1e-53], N[Not[LessEqual[t, 1.1e+55]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.1000000000000001e-53 or 1.10000000000000005e55 < t

   1. Initial program 49.6%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.7%

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

   if -4.1000000000000001e-53 < t < 1.10000000000000005e55

   1. Initial program 89.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.2

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 57.1

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

   10. Applied rewrites 57.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 66.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-53} \lor \neg \left(t \leq 1.1 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.5e+92) y (if (<= t 2.3e+70) (fma (/ z a) y x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.5e+92) {
		tmp = y;
	} else if (t <= 2.3e+70) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.5e+92)
		tmp = y;
	elseif (t <= 2.3e+70)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.5e+92], y, If[LessEqual[t, 2.3e+70], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -1.50000000000000007e92 or 2.29999999999999994e70 < t

   1. Initial program 42.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 50.0%

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

   if -1.50000000000000007e92 < t < 2.29999999999999994e70

   1. Initial program 85.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.5

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

   7. Applied rewrites 58.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 50.5

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

   10. Applied rewrites 50.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 59.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+32}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.8e+32) y (if (<= t 3.8e+51) (fma t (/ x a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.8e+32) {
		tmp = y;
	} else if (t <= 3.8e+51) {
		tmp = fma(t, (x / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.8e+32)
		tmp = y;
	elseif (t <= 3.8e+51)
		tmp = fma(t, Float64(x / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.8e+32], y, If[LessEqual[t, 3.8e+51], N[(t * N[(x / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -4.79999999999999983e32 or 3.7999999999999997e51 < t

   1. Initial program 47.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.5%

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

   if -4.79999999999999983e32 < t < 3.7999999999999997e51

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 41.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 41.3%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+32}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+147}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-61}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.45e+147) x (if (<= a 4.2e-61) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.45e+147) {
		tmp = x;
	} else if (a <= 4.2e-61) {
		tmp = y;
	} else {
		tmp = x;
	}
	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) :: tmp
    if (a <= (-1.45d+147)) then
        tmp = x
    else if (a <= 4.2d-61) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.45e+147) {
		tmp = x;
	} else if (a <= 4.2e-61) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.45e+147:
		tmp = x
	elif a <= 4.2e-61:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.45e+147)
		tmp = x;
	elseif (a <= 4.2e-61)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.45e+147)
		tmp = x;
	elseif (a <= 4.2e-61)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.45e+147], x, If[LessEqual[a, 4.2e-61], y, x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4499999999999999e147 or 4.1999999999999998e-61 < a

   1. Initial program 76.8%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 52.9%

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

   if -1.4499999999999999e147 < a < 4.1999999999999998e-61

   1. Initial program 66.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+147}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-61}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 28.0%

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

6. Final simplification 28.0%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 86.3% 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 2025026 
(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))))