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

Percentage Accurate: 68.5% → 93.1%

Time: 5.3s

Alternatives: 20

Speedup: 0.8×

Specification

Math

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

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: 68.5% accurate, 1.0× speedup

Math

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

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: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ t_2 := \frac{1}{y - x}\\ t_3 := \frac{\frac{x}{y - x} + \frac{t}{t - a}}{t\_2} - \frac{\frac{z}{t - a}}{t\_2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-230}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-299}:\\ \;\;\;\;\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t))))
       (t_2 (/ 1.0 (- y x)))
       (t_3
        (-
         (/ (+ (/ x (- y x)) (/ t (- t a))) t_2)
         (/ (/ z (- t a)) t_2))))
  (if (<= t_1 -1e-230)
    t_3
    (if (<= t_1 1e-299)
      (-
       (+ y (* -1.0 (/ (* z (- y x)) t)))
       (* -1.0 (/ (* a (- y x)) t)))
      t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double t_2 = 1.0 / (y - x);
	double t_3 = (((x / (y - x)) + (t / (t - a))) / t_2) - ((z / (t - a)) / t_2);
	double tmp;
	if (t_1 <= -1e-230) {
		tmp = t_3;
	} else if (t_1 <= 1e-299) {
		tmp = (y + (-1.0 * ((z * (y - x)) / t))) - (-1.0 * ((a * (y - x)) / t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x + (((y - x) * (z - t)) / (a - t))
    t_2 = 1.0d0 / (y - x)
    t_3 = (((x / (y - x)) + (t / (t - a))) / t_2) - ((z / (t - a)) / t_2)
    if (t_1 <= (-1d-230)) then
        tmp = t_3
    else if (t_1 <= 1d-299) then
        tmp = (y + ((-1.0d0) * ((z * (y - x)) / t))) - ((-1.0d0) * ((a * (y - x)) / t))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double t_2 = 1.0 / (y - x);
	double t_3 = (((x / (y - x)) + (t / (t - a))) / t_2) - ((z / (t - a)) / t_2);
	double tmp;
	if (t_1 <= -1e-230) {
		tmp = t_3;
	} else if (t_1 <= 1e-299) {
		tmp = (y + (-1.0 * ((z * (y - x)) / t))) - (-1.0 * ((a * (y - x)) / t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) * (z - t)) / (a - t))
	t_2 = 1.0 / (y - x)
	t_3 = (((x / (y - x)) + (t / (t - a))) / t_2) - ((z / (t - a)) / t_2)
	tmp = 0
	if t_1 <= -1e-230:
		tmp = t_3
	elif t_1 <= 1e-299:
		tmp = (y + (-1.0 * ((z * (y - x)) / t))) - (-1.0 * ((a * (y - x)) / t))
	else:
		tmp = t_3
	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)))
	t_2 = Float64(1.0 / Float64(y - x))
	t_3 = Float64(Float64(Float64(Float64(x / Float64(y - x)) + Float64(t / Float64(t - a))) / t_2) - Float64(Float64(z / Float64(t - a)) / t_2))
	tmp = 0.0
	if (t_1 <= -1e-230)
		tmp = t_3;
	elseif (t_1 <= 1e-299)
		tmp = Float64(Float64(y + Float64(-1.0 * Float64(Float64(z * Float64(y - x)) / t))) - Float64(-1.0 * Float64(Float64(a * Float64(y - x)) / t)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) * (z - t)) / (a - t));
	t_2 = 1.0 / (y - x);
	t_3 = (((x / (y - x)) + (t / (t - a))) / t_2) - ((z / (t - a)) / t_2);
	tmp = 0.0;
	if (t_1 <= -1e-230)
		tmp = t_3;
	elseif (t_1 <= 1e-299)
		tmp = (y + (-1.0 * ((z * (y - x)) / t))) - (-1.0 * ((a * (y - x)) / t));
	else
		tmp = t_3;
	end
	tmp_2 = 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]}, Block[{t$95$2 = N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(x / N[(y - x), $MachinePrecision]), $MachinePrecision] + N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-230], t$95$3, If[LessEqual[t$95$1, 1e-299], N[(N[(y + N[(-1.0 * N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(a * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1e-230 or 9.9999999999999999e-300 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 68.5%

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

   3. Applied rewrites 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 80.9%

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

   5. Applied rewrites 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 84.9%

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

   7. Applied rewrites 84.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

   9. Applied rewrites 88.5%

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

   if -1e-230 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 9.9999999999999999e-300

   1. Initial program 68.5%

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

   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. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 46.0%

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

   4. Applied rewrites 46.0%

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

Alternative 2: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t - a}, z - t, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-299}:\\ \;\;\;\;\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
  (if (<= t_1 -1e-230)
    (fma (/ (- x y) (- t a)) (- z t) x)
    (if (<= t_1 1e-299)
      (-
       (+ y (* -1.0 (/ (* z (- y x)) t)))
       (* -1.0 (/ (* a (- y x)) t)))
      (fma (/ (- t z) (- t a)) (- y x) 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 <= -1e-230) {
		tmp = fma(((x - y) / (t - a)), (z - t), x);
	} else if (t_1 <= 1e-299) {
		tmp = (y + (-1.0 * ((z * (y - x)) / t))) - (-1.0 * ((a * (y - x)) / t));
	} else {
		tmp = fma(((t - z) / (t - a)), (y - x), 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 <= -1e-230)
		tmp = fma(Float64(Float64(x - y) / Float64(t - a)), Float64(z - t), x);
	elseif (t_1 <= 1e-299)
		tmp = Float64(Float64(y + Float64(-1.0 * Float64(Float64(z * Float64(y - x)) / t))) - Float64(-1.0 * Float64(Float64(a * Float64(y - x)) / t)));
	else
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), 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, -1e-230], N[(N[(N[(x - y), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-299], N[(N[(y + N[(-1.0 * N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(a * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $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))) < -1e-230

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.8%

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

   3. Applied rewrites 80.8%

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

   if -1e-230 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 9.9999999999999999e-300

   1. Initial program 68.5%

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

   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. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   if 9.9999999999999999e-300 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

Alternative 3: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t - a}, z - t, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-299}:\\ \;\;\;\;y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
  (if (<= t_1 -1e-230)
    (fma (/ (- x y) (- t a)) (- z t) x)
    (if (<= t_1 1e-299)
      (+ y (* -1.0 (/ (- (* z (- y x)) (* a (- y x))) t)))
      (fma (/ (- t z) (- t a)) (- y x) 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 <= -1e-230) {
		tmp = fma(((x - y) / (t - a)), (z - t), x);
	} else if (t_1 <= 1e-299) {
		tmp = y + (-1.0 * (((z * (y - x)) - (a * (y - x))) / t));
	} else {
		tmp = fma(((t - z) / (t - a)), (y - x), 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 <= -1e-230)
		tmp = fma(Float64(Float64(x - y) / Float64(t - a)), Float64(z - t), x);
	elseif (t_1 <= 1e-299)
		tmp = Float64(y + Float64(-1.0 * Float64(Float64(Float64(z * Float64(y - x)) - Float64(a * Float64(y - x))) / t)));
	else
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), 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, -1e-230], N[(N[(N[(x - y), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-299], N[(y + N[(-1.0 * N[(N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $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))) < -1e-230

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.8%

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

   3. Applied rewrites 80.8%

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

   if -1e-230 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 9.9999999999999999e-300

   1. Initial program 68.5%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 47.0%

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

   4. Applied rewrites 47.0%

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

   if 9.9999999999999999e-300 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

Alternative 4: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t - a}, z - t, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-299}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{t - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
  (if (<= t_1 -1e-230)
    (fma (/ (- x y) (- t a)) (- z t) x)
    (if (<= t_1 1e-299)
      (/ (* y (- t z)) (- t a))
      (fma (/ (- t z) (- t a)) (- y x) 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 <= -1e-230) {
		tmp = fma(((x - y) / (t - a)), (z - t), x);
	} else if (t_1 <= 1e-299) {
		tmp = (y * (t - z)) / (t - a);
	} else {
		tmp = fma(((t - z) / (t - a)), (y - x), 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 <= -1e-230)
		tmp = fma(Float64(Float64(x - y) / Float64(t - a)), Float64(z - t), x);
	elseif (t_1 <= 1e-299)
		tmp = Float64(Float64(y * Float64(t - z)) / Float64(t - a));
	else
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), 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, -1e-230], N[(N[(N[(x - y), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-299], N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $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))) < -1e-230

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.8%

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

   3. Applied rewrites 80.8%

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

   if -1e-230 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 9.9999999999999999e-300

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.5%

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

   9. Applied rewrites 39.5%

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

   if 9.9999999999999999e-300 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

Alternative 5: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-299}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{t - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- t z) (- t a)) (- y x) x))
       (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
  (if (<= t_2 -1e-230)
    t_1
    (if (<= t_2 1e-299) (/ (* y (- t z)) (- t a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / (t - a)), (y - x), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-230) {
		tmp = t_1;
	} else if (t_2 <= 1e-299) {
		tmp = (y * (t - z)) / (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -1e-230)
		tmp = t_1;
	elseif (t_2 <= 1e-299)
		tmp = Float64(Float64(y * Float64(t - z)) / Float64(t - a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $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, -1e-230], t$95$1, If[LessEqual[t$95$2, 1e-299], N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $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))) < -1e-230 or 9.9999999999999999e-300 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   if -1e-230 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 9.9999999999999999e-300

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.5%

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

   9. Applied rewrites 39.5%

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

Alternative 6: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{z}{a - t} - \frac{t}{a - t}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= t 3.5e+146)
  (fma (/ (- t z) (- t a)) (- y x) x)
  (* y (- (/ z (- a t)) (/ t (- a t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.5e+146) {
		tmp = fma(((t - z) / (t - a)), (y - x), x);
	} else {
		tmp = y * ((z / (a - t)) - (t / (a - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.5e+146)
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), x);
	else
		tmp = Float64(y * Float64(Float64(z / Float64(a - t)) - Float64(t / Float64(a - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.5e+146], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5000000000000001e146

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   if 3.5000000000000001e146 < t

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 51.7%

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

   4. Applied rewrites 51.7%

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

Alternative 7: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= y -2.4e-90)
  (+ x (/ (- z t) (/ (- a t) y)))
  (if (<= y 2.8e-56)
    (+ x (/ (* z (- y x)) (- a t)))
    (fma (/ (- t z) (- t a)) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.4e-90) {
		tmp = x + ((z - t) / ((a - t) / y));
	} else if (y <= 2.8e-56) {
		tmp = x + ((z * (y - x)) / (a - t));
	} else {
		tmp = fma(((t - z) / (t - a)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.4e-90)
		tmp = Float64(x + Float64(Float64(z - t) / Float64(Float64(a - t) / y)));
	elseif (y <= 2.8e-56)
		tmp = Float64(x + Float64(Float64(z * Float64(y - x)) / Float64(a - t)));
	else
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.4e-90], N[(x + N[(N[(z - t), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-56], N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4000000000000002e-90

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 68.5%

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

   3. Applied rewrites 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 80.9%

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.5%

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

   if -2.4000000000000002e-90 < y < 2.7999999999999999e-56

   1. Initial program 68.5%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 55.5%

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

   4. Applied rewrites 55.5%

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

   if 2.7999999999999999e-56 < y

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.9%

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

Alternative 8: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t - a}, y, x\right)\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- t z) (- t a)) y x)))
  (if (<= y -2.4e-90)
    t_1
    (if (<= y 2.8e-56) (+ x (/ (* z (- y x)) (- a t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / (t - a)), y, x);
	double tmp;
	if (y <= -2.4e-90) {
		tmp = t_1;
	} else if (y <= 2.8e-56) {
		tmp = x + ((z * (y - x)) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / Float64(t - a)), y, x)
	tmp = 0.0
	if (y <= -2.4e-90)
		tmp = t_1;
	elseif (y <= 2.8e-56)
		tmp = Float64(x + Float64(Float64(z * Float64(y - x)) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[y, -2.4e-90], t$95$1, If[LessEqual[y, 2.8e-56], N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4000000000000002e-90 or 2.7999999999999999e-56 < y

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.9%

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

   if -2.4000000000000002e-90 < y < 2.7999999999999999e-56

   1. Initial program 68.5%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 55.5%

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

   4. Applied rewrites 55.5%

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

Alternative 9: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t - a}, y, x\right)\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t - a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- t z) (- t a)) y x)))
  (if (<= y -1.4e-34)
    t_1
    (if (<= y 3.1e-96) (fma (/ x (- t a)) (- z t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / (t - a)), y, x);
	double tmp;
	if (y <= -1.4e-34) {
		tmp = t_1;
	} else if (y <= 3.1e-96) {
		tmp = fma((x / (t - a)), (z - t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / Float64(t - a)), y, x)
	tmp = 0.0
	if (y <= -1.4e-34)
		tmp = t_1;
	elseif (y <= 3.1e-96)
		tmp = fma(Float64(x / Float64(t - a)), Float64(z - t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[y, -1.4e-34], t$95$1, If[LessEqual[y, 3.1e-96], N[(N[(x / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e-34 or 3.0999999999999999e-96 < y

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.9%

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

   if -1.4e-34 < y < 3.0999999999999999e-96

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.8%

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 42.1%

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

   6. Applied rewrites 42.1%

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

Alternative 10: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(1 - \frac{z}{t}, y - x, x\right)\\ \mathbf{if}\;t \leq -21000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{y - x}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (- 1.0 (/ z t)) (- y x) x)))
  (if (<= t -21000000000.0)
    t_1
    (if (<= t 1.25e+14) (+ x (* (/ (- y x) a) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((1.0 - (z / t)), (y - x), x);
	double tmp;
	if (t <= -21000000000.0) {
		tmp = t_1;
	} else if (t <= 1.25e+14) {
		tmp = x + (((y - x) / a) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(1.0 - Float64(z / t)), Float64(y - x), x)
	tmp = 0.0
	if (t <= -21000000000.0)
		tmp = t_1;
	elseif (t <= 1.25e+14)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / a) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -21000000000.0], t$95$1, If[LessEqual[t, 1.25e+14], N[(x + N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1e10 or 1.25e14 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 40.0%

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

   8. Applied rewrites 40.0%

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

   if -2.1e10 < t < 1.25e14

   1. Initial program 68.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 44.5%

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

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3%

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

   6. Applied rewrites 48.3%

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

Alternative 11: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 11000:\\ \;\;\;\;x + \frac{y - x}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- t z) t) y x)))
  (if (<= t -2.2e+18)
    t_1
    (if (<= t 11000.0) (+ x (* (/ (- y x) a) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / t), y, x);
	double tmp;
	if (t <= -2.2e+18) {
		tmp = t_1;
	} else if (t <= 11000.0) {
		tmp = x + (((y - x) / a) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / t), y, x)
	tmp = 0.0
	if (t <= -2.2e+18)
		tmp = t_1;
	elseif (t <= 11000.0)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / a) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.2e+18], t$95$1, If[LessEqual[t, 11000.0], N[(x + N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2e18 or 11000 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 43.5%

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

   if -2.2e18 < t < 11000

   1. Initial program 68.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 44.5%

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

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3%

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

   6. Applied rewrites 48.3%

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

Alternative 12: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 11000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- t z) t) y x)))
  (if (<= t -2.2e+18)
    t_1
    (if (<= t 11000.0) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / t), y, x);
	double tmp;
	if (t <= -2.2e+18) {
		tmp = t_1;
	} else if (t <= 11000.0) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / t), y, x)
	tmp = 0.0
	if (t <= -2.2e+18)
		tmp = t_1;
	elseif (t <= 11000.0)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.2e+18], t$95$1, If[LessEqual[t, 11000.0], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2e18 or 11000 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 43.5%

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

   if -2.2e18 < t < 11000

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 49.5%

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

   6. Applied rewrites 49.5%

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

Alternative 13: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- t z) t) y x)))
  (if (<= t -2400.0) t_1 (if (<= t 1.5e-16) (+ x (* (/ y a) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / t), y, x);
	double tmp;
	if (t <= -2400.0) {
		tmp = t_1;
	} else if (t <= 1.5e-16) {
		tmp = x + ((y / a) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / t), y, x)
	tmp = 0.0
	if (t <= -2400.0)
		tmp = t_1;
	elseif (t <= 1.5e-16)
		tmp = Float64(x + Float64(Float64(y / a) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2400.0], t$95$1, If[LessEqual[t, 1.5e-16], N[(x + N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2400 or 1.5e-16 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 43.5%

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

   if -2400 < t < 1.5e-16

   1. Initial program 68.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 44.5%

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

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3%

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 40.1%

      \[\leadsto x + \frac{y}{a} \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.95:\\ \;\;\;\;\frac{x \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 3100:\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma 1.0 (- y x) x)))
  (if (<= t -1.4e+126)
    t_1
    (if (<= t -2.95)
      (/ (* x (- z a)) t)
      (if (<= t 3100.0) (+ x (* (/ y a) z)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -1.4e+126) {
		tmp = t_1;
	} else if (t <= -2.95) {
		tmp = (x * (z - a)) / t;
	} else if (t <= 3100.0) {
		tmp = x + ((y / a) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -1.4e+126)
		tmp = t_1;
	elseif (t <= -2.95)
		tmp = Float64(Float64(x * Float64(z - a)) / t);
	elseif (t <= 3100.0)
		tmp = Float64(x + Float64(Float64(y / a) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.4e+126], t$95$1, If[LessEqual[t, -2.95], N[(N[(x * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 3100.0], N[(x + N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.4e126 or 3100 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 19.8%

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

   if -1.4e126 < t < -2.9500000000000002

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 25.4%

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 20.2%

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

   13. Applied rewrites 20.2%

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

   if -2.9500000000000002 < t < 3100

   1. Initial program 68.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 44.5%

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

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3%

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 40.1%

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

Alternative 15: 35.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.95:\\ \;\;\;\;\frac{x \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 + \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma 1.0 (- y x) x)))
  (if (<= t -1.4e+126)
    t_1
    (if (<= t -2.95)
      (/ (* x (- z a)) t)
      (if (<= t 5.8e-5) (* x (+ 1.0 (/ t a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -1.4e+126) {
		tmp = t_1;
	} else if (t <= -2.95) {
		tmp = (x * (z - a)) / t;
	} else if (t <= 5.8e-5) {
		tmp = x * (1.0 + (t / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -1.4e+126)
		tmp = t_1;
	elseif (t <= -2.95)
		tmp = Float64(Float64(x * Float64(z - a)) / t);
	elseif (t <= 5.8e-5)
		tmp = Float64(x * Float64(1.0 + Float64(t / a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.4e+126], t$95$1, If[LessEqual[t, -2.95], N[(N[(x * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 5.8e-5], N[(x * N[(1.0 + N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.4e126 or 5.8e-5 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 19.8%

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

   if -1.4e126 < t < -2.9500000000000002

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 25.4%

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 20.2%

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

   13. Applied rewrites 20.2%

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

   if -2.9500000000000002 < t < 5.8e-5

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 25.4%

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 26.2%

       \[\leadsto x \cdot \left(1 + \frac{t}{a}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 35.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.95:\\ \;\;\;\;\frac{x \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 10000000000000:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma 1.0 (- y x) x)))
  (if (<= t -1.4e+126)
    t_1
    (if (<= t -2.95)
      (/ (* x (- z a)) t)
      (if (<= t 10000000000000.0) (* x 1.0) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -1.4e+126) {
		tmp = t_1;
	} else if (t <= -2.95) {
		tmp = (x * (z - a)) / t;
	} else if (t <= 10000000000000.0) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -1.4e+126)
		tmp = t_1;
	elseif (t <= -2.95)
		tmp = Float64(Float64(x * Float64(z - a)) / t);
	elseif (t <= 10000000000000.0)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.4e+126], t$95$1, If[LessEqual[t, -2.95], N[(N[(x * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 10000000000000.0], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.4e126 or 1e13 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 19.8%

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

   if -1.4e126 < t < -2.9500000000000002

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 25.4%

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 20.2%

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

   13. Applied rewrites 20.2%

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

   if -2.9500000000000002 < t < 1e13

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 25.4%

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around 0

       \[\leadsto x \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 25.1%

       \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 34.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.95:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 10000000000000:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma 1.0 (- y x) x)))
  (if (<= t -6.8e+107)
    t_1
    (if (<= t -2.95)
      (/ (* x z) t)
      (if (<= t 10000000000000.0) (* x 1.0) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -6.8e+107) {
		tmp = t_1;
	} else if (t <= -2.95) {
		tmp = (x * z) / t;
	} else if (t <= 10000000000000.0) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -6.8e+107)
		tmp = t_1;
	elseif (t <= -2.95)
		tmp = Float64(Float64(x * z) / t);
	elseif (t <= 10000000000000.0)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -6.8e+107], t$95$1, If[LessEqual[t, -2.95], N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 10000000000000.0], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.7999999999999994e107 or 1e13 < t

   1. Initial program 68.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 85.0%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 19.8%

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

   if -6.7999999999999994e107 < t < -2.9500000000000002

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.9%

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

   7. Applied rewrites 16.9%

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

   if -2.9500000000000002 < t < 1e13

   1. Initial program 68.5%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{t}{a - \color{blue}{t}}\right) \]

      4. lower--.f64 25.4%

         \[\leadsto x \cdot \left(1 + \frac{t}{a - t}\right) \]

   7. Applied rewrites 25.4%

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around 0

       \[\leadsto x \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 25.1%

       \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 34.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-52}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{z}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= a -1.02e-52)
  (* x 1.0)
  (if (<= a 2e-30) (* (/ z t) x) (* x 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.02e-52) {
		tmp = x * 1.0;
	} else if (a <= 2e-30) {
		tmp = (z / t) * x;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.02d-52)) then
        tmp = x * 1.0d0
    else if (a <= 2d-30) then
        tmp = (z / t) * x
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.02e-52) {
		tmp = x * 1.0;
	} else if (a <= 2e-30) {
		tmp = (z / t) * x;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.02e-52:
		tmp = x * 1.0
	elif a <= 2e-30:
		tmp = (z / t) * x
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.02e-52)
		tmp = Float64(x * 1.0);
	elseif (a <= 2e-30)
		tmp = Float64(Float64(z / t) * x);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.02e-52)
		tmp = x * 1.0;
	elseif (a <= 2e-30)
		tmp = (z / t) * x;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.02e-52], N[(x * 1.0), $MachinePrecision], If[LessEqual[a, 2e-30], N[(N[(z / t), $MachinePrecision] * x), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0200000000000001e-52 or 2.0000000000000002e-30 < a

   1. Initial program 68.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)}\right) \]

      3. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \color{blue}{\left(1 + \frac{t}{a - t}\right)}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(\color{blue}{1} + \frac{t}{a - t}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \color{blue}{\frac{t}{a - t}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{\color{blue}{a - t}}\right)\right)\right) \]

      8. lower--.f64 47.5%

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - \color{blue}{t}}\right)\right)\right) \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{t}{a - t}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{t}{\color{blue}{a - t}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{t}{a - \color{blue}{t}}\right) \]

      4. lower--.f64 25.4%

         \[\leadsto x \cdot \left(1 + \frac{t}{a - t}\right) \]

   7. Applied rewrites 25.4%

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around 0

       \[\leadsto x \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 25.1%

       \[\leadsto x \cdot 1 \]

   if -1.0200000000000001e-52 < a < 2.0000000000000002e-30

   1. Initial program 68.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)}\right) \]

      3. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \color{blue}{\left(1 + \frac{t}{a - t}\right)}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(\color{blue}{1} + \frac{t}{a - t}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \color{blue}{\frac{t}{a - t}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{\color{blue}{a - t}}\right)\right)\right) \]

      8. lower--.f64 47.5%

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - \color{blue}{t}}\right)\right)\right) \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{x \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot z}{t} \]

      2. lower-*.f64 16.9%

         \[\leadsto \frac{x \cdot z}{t} \]

   7. Applied rewrites 16.9%

      \[\leadsto \frac{x \cdot z}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot z}{t} \]

      3. associate-/l* N/A

         \[\leadsto x \cdot \frac{z}{\color{blue}{t}} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{t} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z}{t} \cdot x \]

      6. lower-/.f64 19.2%

         \[\leadsto \frac{z}{t} \cdot x \]

   9. Applied rewrites 19.2%

      \[\leadsto \frac{z}{t} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 34.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-52}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-30}:\\ \;\;\;\;z \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= a -1.02e-52)
  (* x 1.0)
  (if (<= a 2e-30) (* z (/ x t)) (* x 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.02e-52) {
		tmp = x * 1.0;
	} else if (a <= 2e-30) {
		tmp = z * (x / t);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.02d-52)) then
        tmp = x * 1.0d0
    else if (a <= 2d-30) then
        tmp = z * (x / t)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.02e-52) {
		tmp = x * 1.0;
	} else if (a <= 2e-30) {
		tmp = z * (x / t);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.02e-52:
		tmp = x * 1.0
	elif a <= 2e-30:
		tmp = z * (x / t)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.02e-52)
		tmp = Float64(x * 1.0);
	elseif (a <= 2e-30)
		tmp = Float64(z * Float64(x / t));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.02e-52)
		tmp = x * 1.0;
	elseif (a <= 2e-30)
		tmp = z * (x / t);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.02e-52], N[(x * 1.0), $MachinePrecision], If[LessEqual[a, 2e-30], N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0200000000000001e-52 or 2.0000000000000002e-30 < a

   1. Initial program 68.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)}\right) \]

      3. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \color{blue}{\left(1 + \frac{t}{a - t}\right)}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(\color{blue}{1} + \frac{t}{a - t}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \color{blue}{\frac{t}{a - t}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{\color{blue}{a - t}}\right)\right)\right) \]

      8. lower--.f64 47.5%

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - \color{blue}{t}}\right)\right)\right) \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{t}{a - t}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{t}{\color{blue}{a - t}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{t}{a - \color{blue}{t}}\right) \]

      4. lower--.f64 25.4%

         \[\leadsto x \cdot \left(1 + \frac{t}{a - t}\right) \]

   7. Applied rewrites 25.4%

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 2.8%

       \[\leadsto x \cdot \left(1 + -1\right) \]

   11. Taylor expanded in t around 0

       \[\leadsto x \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 25.1%

       \[\leadsto x \cdot 1 \]

   if -1.0200000000000001e-52 < a < 2.0000000000000002e-30

   1. Initial program 68.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)}\right) \]

      3. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \color{blue}{\left(1 + \frac{t}{a - t}\right)}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(\color{blue}{1} + \frac{t}{a - t}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \color{blue}{\frac{t}{a - t}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{\color{blue}{a - t}}\right)\right)\right) \]

      8. lower--.f64 47.5%

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - \color{blue}{t}}\right)\right)\right) \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{x \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot z}{t} \]

      2. lower-*.f64 16.9%

         \[\leadsto \frac{x \cdot z}{t} \]

   7. Applied rewrites 16.9%

      \[\leadsto \frac{x \cdot z}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot z}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{z \cdot x}{t} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{x}{\color{blue}{t}} \]

      5. frac-2neg-rev N/A

         \[\leadsto z \cdot \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(t\right)} \]

      6. lift-neg.f64 N/A

         \[\leadsto z \cdot \frac{-x}{\mathsf{neg}\left(t\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto z \cdot \frac{-x}{\color{blue}{\mathsf{neg}\left(t\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto z \cdot \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(t\right)} \]

      9. frac-2neg-rev N/A

         \[\leadsto z \cdot \frac{x}{t} \]

      10. lower-/.f64 18.1%

          \[\leadsto z \cdot \frac{x}{t} \]

   9. Applied rewrites 18.1%

      \[\leadsto z \cdot \frac{x}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.1% accurate, 4.5× speedup

Math

\[x \cdot 1 \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (* x 1.0))

C

double code(double x, double y, double z, double t, double a) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x * 1.0;
}

Python

def code(x, y, z, t, a):
	return x * 1.0

Julia

function code(x, y, z, t, a)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

2. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)}\right) \]

   3. lower--.f64 N/A

      \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \color{blue}{\left(1 + \frac{t}{a - t}\right)}\right)\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(\color{blue}{1} + \frac{t}{a - t}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \color{blue}{\frac{t}{a - t}}\right)\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{\color{blue}{a - t}}\right)\right)\right) \]

   8. lower--.f64 47.5%

      \[\leadsto -1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - \color{blue}{t}}\right)\right)\right) \]

4. Applied rewrites 47.5%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{t}{a - t}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto x \cdot \left(1 + \frac{t}{\color{blue}{a - t}}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto x \cdot \left(1 + \frac{t}{a - \color{blue}{t}}\right) \]

   4. lower--.f64 25.4%

      \[\leadsto x \cdot \left(1 + \frac{t}{a - t}\right) \]

7. Applied rewrites 25.4%

   \[\leadsto x \cdot \color{blue}{\left(1 + \frac{t}{a - t}\right)} \]

8. Taylor expanded in t around inf

   \[\leadsto x \cdot \left(1 + -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 2.8%

    \[\leadsto x \cdot \left(1 + -1\right) \]

11. Taylor expanded in t around 0

    \[\leadsto x \cdot 1 \]
12. Step-by-step derivation

13. Applied rewrites 25.1%

    \[\leadsto x \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))