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

Percentage Accurate: 68.8% → 91.2%

Time: 4.8s

Alternatives: 17

Speedup: 0.8×

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: 68.8% 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: 91.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.8%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
2. 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. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. sub-div N/A

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

   10. lower-fma.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lift--.f64 83.7

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

3. Applied rewrites 83.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-div N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift--.f64 83.7

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in x around -inf

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. sub-div N/A

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

8. Applied rewrites 82.8%

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

9. Taylor expanded in y around 0

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

    1. sub-div N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift--.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. lift--.f64 N/A

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

    9. lift-+.f64 N/A

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

11. Applied rewrites 91.2%

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

Alternative 2: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{a - t}\\ t_2 := \frac{t}{a - t}\\ t_3 := -1 \cdot \left(x \cdot \left(\mathsf{fma}\left(-1, \frac{y}{x}, t\_1\right) - \left(1 + t\_2\right)\right)\right)\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+110}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y - x, t\_1 - t\_2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ z (- a t)))
        (t_2 (/ t (- a t)))
        (t_3 (* -1.0 (* x (- (fma -1.0 (/ y x) t_1) (+ 1.0 t_2))))))
   (if (<= x -8.8e+110)
     t_3
     (if (<= x 1.75e+44) (fma (- y x) (- t_1 t_2) x) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z / (a - t);
	double t_2 = t / (a - t);
	double t_3 = -1.0 * (x * (fma(-1.0, (y / x), t_1) - (1.0 + t_2)));
	double tmp;
	if (x <= -8.8e+110) {
		tmp = t_3;
	} else if (x <= 1.75e+44) {
		tmp = fma((y - x), (t_1 - t_2), x);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z / Float64(a - t))
	t_2 = Float64(t / Float64(a - t))
	t_3 = Float64(-1.0 * Float64(x * Float64(fma(-1.0, Float64(y / x), t_1) - Float64(1.0 + t_2))))
	tmp = 0.0
	if (x <= -8.8e+110)
		tmp = t_3;
	elseif (x <= 1.75e+44)
		tmp = fma(Float64(y - x), Float64(t_1 - t_2), x);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * N[(x * N[(N[(-1.0 * N[(y / x), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.8e+110], t$95$3, If[LessEqual[x, 1.75e+44], N[(N[(y - x), $MachinePrecision] * N[(t$95$1 - t$95$2), $MachinePrecision] + x), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.79999999999999967e110 or 1.75e44 < x

   1. Initial program 55.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around -inf

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sub-div N/A

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

   8. Applied rewrites 86.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 77.0%

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

   if -8.79999999999999967e110 < x < 1.75e44

   1. Initial program 76.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.4

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

   3. Applied rewrites 89.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 89.4

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

   5. Applied rewrites 89.4%

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

Alternative 3: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_1 -2e-209)
     (fma (- y x) (- (/ z (- a t)) (/ t (- a t))) x)
     (if (<= t_1 0.0)
       (+ (- (/ (* (- y x) (- z a)) t)) y)
       (fma (- y x) (/ (- z t) (- a t)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -2e-209) {
		tmp = fma((y - x), ((z / (a - t)) - (t / (a - t))), x);
	} else if (t_1 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -2e-209)
		tmp = fma(Float64(y - x), Float64(Float64(z / Float64(a - t)) - Float64(t / Float64(a - t))), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-209], N[(N[(y - x), $MachinePrecision] * N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2.0000000000000001e-209

   1. Initial program 73.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 90.1

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

   3. Applied rewrites 90.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if -2.0000000000000001e-209 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 86.7%

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

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

   1. Initial program 74.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.9

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

   3. Applied rewrites 89.9%

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

Alternative 4: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) (- a t)) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -2e-209)
     t_1
     (if (<= t_2 0.0) (+ (- (/ (* (- y x) (- z a)) t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / (a - t)), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -2e-209) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e-209)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-209], t$95$1, If[LessEqual[t$95$2, 0.0], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2.0000000000000001e-209 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 73.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 90.0

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

   3. Applied rewrites 90.0%

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

   if -2.0000000000000001e-209 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 86.7%

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

Alternative 5: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+99}:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.5e+99)
   (+ (- (/ (* (- y x) (- z a)) t)) y)
   (if (<= t 5e+40) (fma (- y x) (/ z (- a t)) x) (* y (/ (- z t) (- a t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.5e+99) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else if (t <= 5e+40) {
		tmp = fma((y - x), (z / (a - t)), x);
	} else {
		tmp = y * ((z - t) / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.5e+99)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	elseif (t <= 5e+40)
		tmp = fma(Float64(y - x), Float64(z / Float64(a - t)), x);
	else
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.5e+99], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], If[LessEqual[t, 5e+40], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5e99

   1. Initial program 38.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 63.3%

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

   if -4.5e99 < t < 5.00000000000000003e40

   1. Initial program 86.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 80.6%

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

   if 5.00000000000000003e40 < t

   1. Initial program 41.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 68.3

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 59.8

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

   6. Applied rewrites 59.8%

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

Alternative 6: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t -7e+118)
     t_1
     (if (<= t 5e+40) (fma (- y x) (/ z (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t <= -7e+118) {
		tmp = t_1;
	} else if (t <= 5e+40) {
		tmp = fma((y - x), (z / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t <= -7e+118)
		tmp = t_1;
	elseif (t <= 5e+40)
		tmp = fma(Float64(y - x), Float64(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[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7e+118], t$95$1, If[LessEqual[t, 5e+40], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.00000000000000033e118 or 5.00000000000000003e40 < t

   1. Initial program 39.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 68.7

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

   3. Applied rewrites 68.7%

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

   4. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 62.2

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

   6. Applied rewrites 62.2%

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

   if -7.00000000000000033e118 < t < 5.00000000000000003e40

   1. Initial program 85.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.3

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 79.6%

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

Alternative 7: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t -3.7e+114)
     t_1
     (if (<= t 3.6e+36) (fma (- y x) (/ (- z t) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t <= -3.7e+114) {
		tmp = t_1;
	} else if (t <= 3.6e+36) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t <= -3.7e+114)
		tmp = t_1;
	elseif (t <= 3.6e+36)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.7e+114], t$95$1, If[LessEqual[t, 3.6e+36], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7000000000000001e114 or 3.5999999999999997e36 < t

   1. Initial program 40.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 69.0

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

   3. Applied rewrites 69.0%

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

   4. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 62.0

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

   6. Applied rewrites 62.0%

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

   if -3.7000000000000001e114 < t < 3.5999999999999997e36

   1. Initial program 86.0%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

Alternative 8: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t -3.5e-71) t_1 (if (<= t 7.2e-10) (fma (- y x) (/ z a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t <= -3.5e-71) {
		tmp = t_1;
	} else if (t <= 7.2e-10) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t <= -3.5e-71)
		tmp = t_1;
	elseif (t <= 7.2e-10)
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.5e-71], t$95$1, If[LessEqual[t, 7.2e-10], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.4999999999999999e-71 or 7.2e-10 < t

   1. Initial program 51.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 74.7

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if -3.4999999999999999e-71 < t < 7.2e-10

   1. Initial program 90.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 95.2

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

   3. Applied rewrites 95.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 77.6

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

   6. Applied rewrites 77.6%

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

Alternative 9: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+244}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -5.2e+244)
   y
   (if (<= t -1.5e+173)
     (* (/ (- z a) t) x)
     (if (<= t -3.5e-71)
       (/ (* (- z t) y) (- a t))
       (if (<= t 2.4e+37) (fma (- y x) (/ z a) x) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -5.2e+244) {
		tmp = y;
	} else if (t <= -1.5e+173) {
		tmp = ((z - a) / t) * x;
	} else if (t <= -3.5e-71) {
		tmp = ((z - t) * y) / (a - t);
	} else if (t <= 2.4e+37) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -5.2e+244)
		tmp = y;
	elseif (t <= -1.5e+173)
		tmp = Float64(Float64(Float64(z - a) / t) * x);
	elseif (t <= -3.5e-71)
		tmp = Float64(Float64(Float64(z - t) * y) / Float64(a - t));
	elseif (t <= 2.4e+37)
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -5.2e+244], y, If[LessEqual[t, -1.5e+173], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, -3.5e-71], N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e+37], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], y]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.2000000000000001e244 or 2.4e37 < t

   1. Initial program 38.4%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 48.6%

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

   if -5.2000000000000001e244 < t < -1.4999999999999999e173

   1. Initial program 39.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 21.8

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

   4. Applied rewrites 21.8%

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

   5. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 26.6

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

   7. Applied rewrites 26.6%

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

   if -1.4999999999999999e173 < t < -3.4999999999999999e-71

   1. Initial program 67.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 42.9

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

   4. Applied rewrites 42.9%

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

   if -3.4999999999999999e-71 < t < 2.4e37

   1. Initial program 89.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 94.4

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

   3. Applied rewrites 94.4%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 74.4

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

   6. Applied rewrites 74.4%

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

Alternative 10: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+119}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.15e+119)
   y
   (if (<= t -2.7e-17)
     (/ (* (- y x) z) (- a t))
     (if (<= t 2.4e+37) (fma (- y x) (/ z a) x) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.15e+119) {
		tmp = y;
	} else if (t <= -2.7e-17) {
		tmp = ((y - x) * z) / (a - t);
	} else if (t <= 2.4e+37) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.15e+119)
		tmp = y;
	elseif (t <= -2.7e-17)
		tmp = Float64(Float64(Float64(y - x) * z) / Float64(a - t));
	elseif (t <= 2.4e+37)
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.15e+119], y, If[LessEqual[t, -2.7e-17], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e+37], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15e119 or 2.4e37 < t

   1. Initial program 39.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 47.8%

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

   if -1.15e119 < t < -2.7000000000000001e-17

   1. Initial program 68.2%

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

   2. Taylor expanded in z around inf

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

      1. sub-div 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. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 34.3

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

   4. Applied rewrites 34.3%

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

   if -2.7000000000000001e-17 < t < 2.4e37

   1. Initial program 89.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 94.2

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 72.8

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

   6. Applied rewrites 72.8%

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

Alternative 11: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+118}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.6e+118) y (if (<= t 2.4e+37) (fma (- y x) (/ z a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.6e+118) {
		tmp = y;
	} else if (t <= 2.4e+37) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.6e+118)
		tmp = y;
	elseif (t <= 2.4e+37)
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.6e+118], y, If[LessEqual[t, 2.4e+37], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -7.60000000000000033e118 or 2.4e37 < t

   1. Initial program 39.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 47.8%

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

   if -7.60000000000000033e118 < t < 2.4e37

   1. Initial program 85.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.3

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 67.1

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

   6. Applied rewrites 67.1%

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

Alternative 12: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+118}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.6e+118) y (if (<= t 2.4e+37) (fma z (/ (- y x) a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.6e+118) {
		tmp = y;
	} else if (t <= 2.4e+37) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.6e+118)
		tmp = y;
	elseif (t <= 2.4e+37)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.6e+118], y, If[LessEqual[t, 2.4e+37], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -7.60000000000000033e118 or 2.4e37 < t

   1. Initial program 39.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 47.8%

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

   if -7.60000000000000033e118 < t < 2.4e37

   1. Initial program 85.8%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   3. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 65.1

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

   4. Applied rewrites 65.1%

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

Alternative 13: 52.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+115}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-297}:\\ \;\;\;\;\left(1 - \frac{z}{a}\right) \cdot x\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.5e+115)
   y
   (if (<= t 2.1e-297)
     (* (- 1.0 (/ z a)) x)
     (if (<= t 1.05e+44) (fma z (/ y a) x) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.5e+115) {
		tmp = y;
	} else if (t <= 2.1e-297) {
		tmp = (1.0 - (z / a)) * x;
	} else if (t <= 1.05e+44) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.5e+115)
		tmp = y;
	elseif (t <= 2.1e-297)
		tmp = Float64(Float64(1.0 - Float64(z / a)) * x);
	elseif (t <= 1.05e+44)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.5e+115], y, If[LessEqual[t, 2.1e-297], N[(N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1.05e+44], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.50000000000000057e115 or 1.04999999999999993e44 < t

   1. Initial program 39.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 48.2%

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

   if -8.50000000000000057e115 < t < 2.10000000000000013e-297

   1. Initial program 83.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 49.2

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

   7. Applied rewrites 49.2%

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

   if 2.10000000000000013e-297 < t < 1.04999999999999993e44

   1. Initial program 88.7%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   3. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 55.5%

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

Alternative 14: 50.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+118}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.6e+118) y (if (<= t 1.05e+44) (fma z (/ y a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.6e+118) {
		tmp = y;
	} else if (t <= 1.05e+44) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.6e+118)
		tmp = y;
	elseif (t <= 1.05e+44)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.6e+118], y, If[LessEqual[t, 1.05e+44], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -7.60000000000000033e118 or 1.04999999999999993e44 < t

   1. Initial program 39.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 48.2%

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

   if -7.60000000000000033e118 < t < 1.04999999999999993e44

   1. Initial program 85.7%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   3. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 54.5%

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

Alternative 15: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+191}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{z}{t} \cdot x\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.4e+191)
   y
   (if (<= t -3.8e-180)
     (* (/ z t) x)
     (if (<= t 1.4e-307) (* y (/ z a)) (if (<= t 6e+40) x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.4e+191) {
		tmp = y;
	} else if (t <= -3.8e-180) {
		tmp = (z / t) * x;
	} else if (t <= 1.4e-307) {
		tmp = y * (z / a);
	} else if (t <= 6e+40) {
		tmp = x;
	} else {
		tmp = y;
	}
	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 (t <= (-4.4d+191)) then
        tmp = y
    else if (t <= (-3.8d-180)) then
        tmp = (z / t) * x
    else if (t <= 1.4d-307) then
        tmp = y * (z / a)
    else if (t <= 6d+40) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.4e+191) {
		tmp = y;
	} else if (t <= -3.8e-180) {
		tmp = (z / t) * x;
	} else if (t <= 1.4e-307) {
		tmp = y * (z / a);
	} else if (t <= 6e+40) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -4.4e+191:
		tmp = y
	elif t <= -3.8e-180:
		tmp = (z / t) * x
	elif t <= 1.4e-307:
		tmp = y * (z / a)
	elif t <= 6e+40:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.4e+191)
		tmp = y;
	elseif (t <= -3.8e-180)
		tmp = Float64(Float64(z / t) * x);
	elseif (t <= 1.4e-307)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 6e+40)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -4.4e+191)
		tmp = y;
	elseif (t <= -3.8e-180)
		tmp = (z / t) * x;
	elseif (t <= 1.4e-307)
		tmp = y * (z / a);
	elseif (t <= 6e+40)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.4e+191], y, If[LessEqual[t, -3.8e-180], N[(N[(z / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1.4e-307], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+40], x, y]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.4e191 or 6.0000000000000004e40 < t

   1. Initial program 37.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 49.4%

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

   if -4.4e191 < t < -3.79999999999999999e-180

   1. Initial program 73.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 19.3

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

   7. Applied rewrites 19.3%

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

   if -3.79999999999999999e-180 < t < 1.4e-307

   1. Initial program 92.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 97.3

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 36.5

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

   6. Applied rewrites 36.5%

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

   7. Taylor expanded in t around 0

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

      1. lift-/.f64 32.9

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

   9. Applied rewrites 32.9%

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

   if 1.4e-307 < t < 6.0000000000000004e40

   1. Initial program 88.9%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 33.0%

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

Alternative 16: 33.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+118}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7e+118) y (if (<= t 6e+40) x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7e+118) {
		tmp = y;
	} else if (t <= 6e+40) {
		tmp = x;
	} else {
		tmp = y;
	}
	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 (t <= (-7d+118)) then
        tmp = y
    else if (t <= 6d+40) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7e+118) {
		tmp = y;
	} else if (t <= 6e+40) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -7e+118:
		tmp = y
	elif t <= 6e+40:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7e+118)
		tmp = y;
	elseif (t <= 6e+40)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -7e+118)
		tmp = y;
	elseif (t <= 6e+40)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7e+118], y, If[LessEqual[t, 6e+40], x, y]]

Derivation

1. Split input into 2 regimes

2. if t < -7.00000000000000033e118 or 6.0000000000000004e40 < t

   1. Initial program 39.4%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 48.0%

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

   if -7.00000000000000033e118 < t < 6.0000000000000004e40

   1. Initial program 85.7%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 33.3%

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

Alternative 17: 25.7% accurate, 17.9× 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 68.8%

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

2. Taylor expanded in a around inf

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

4. Applied rewrites 25.7%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y x) (- z t)) (- a t))))