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

Percentage Accurate: 68.1% → 87.8%

Time: 4.7s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{t}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot z, -1, y\right) - \left(-a \cdot t\_1\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + -1 \cdot \left(x \cdot \frac{a - z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y x) t)))
   (if (<= t -3.6e+197)
     (- (fma (* t_1 z) -1.0 y) (- (* a t_1)))
     (if (<= t 6.8e+165)
       (fma (- y x) (/ (- z t) (- a t)) x)
       (+ y (* -1.0 (* x (/ (- a z) t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) / t;
	double tmp;
	if (t <= -3.6e+197) {
		tmp = fma((t_1 * z), -1.0, y) - -(a * t_1);
	} else if (t <= 6.8e+165) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = y + (-1.0 * (x * ((a - z) / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) / t)
	tmp = 0.0
	if (t <= -3.6e+197)
		tmp = Float64(fma(Float64(t_1 * z), -1.0, y) - Float64(-Float64(a * t_1)));
	elseif (t <= 6.8e+165)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = Float64(y + Float64(-1.0 * Float64(x * Float64(Float64(a - z) / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -3.6e+197], N[(N[(N[(t$95$1 * z), $MachinePrecision] * -1.0 + y), $MachinePrecision] - (-N[(a * t$95$1), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 6.8e+165], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(-1.0 * N[(x * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.59999999999999982e197

   1. Initial program 28.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 66.4

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 84.4

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

   6. Applied rewrites 84.4%

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

   if -3.59999999999999982e197 < t < 6.80000000000000022e165

   1. Initial program 78.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.1

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

   3. Applied rewrites 89.1%

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

   if 6.80000000000000022e165 < t

   1. Initial program 30.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 65.2

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

   3. Applied rewrites 65.2%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 50.2

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

   6. Applied rewrites 50.2%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 66.0

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

   9. Applied rewrites 66.0%

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

   10. Taylor expanded in x around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. sub-div N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 81.0

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

   12. Applied rewrites 81.0%

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

Alternative 2: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-263}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + \frac{x \cdot \left(z - a\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) (- a t)) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -1e-263) t_1 (if (<= t_2 0.0) (+ y (/ (* x (- z a)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / (a - t)), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-263) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y + ((x * (z - a)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -1e-263)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(Float64(x * Float64(z - a)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-263], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y + N[(N[(x * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $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-263 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 72.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 90.3

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

   3. Applied rewrites 90.3%

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

   if -1e-263 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 9.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 9.2

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

   3. Applied rewrites 9.2%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 9.0

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

   6. Applied rewrites 9.0%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 95.1

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

   9. Applied rewrites 95.1%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 95.1

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

   12. Applied rewrites 95.1%

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

Alternative 3: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq -2.65 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-19}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.25e+54)
   (fma y (/ (- z t) a) x)
   (if (<= a -2.65e-130)
     (* y (/ (- z t) (- a t)))
     (if (<= a 1.1e-19) (- y (/ (* z (- y x)) t)) (fma (- y x) (/ z a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.25e+54) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= -2.65e-130) {
		tmp = y * ((z - t) / (a - t));
	} else if (a <= 1.1e-19) {
		tmp = y - ((z * (y - x)) / t);
	} else {
		tmp = fma((y - x), (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.25e+54)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= -2.65e-130)
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	elseif (a <= 1.1e-19)
		tmp = Float64(y - Float64(Float64(z * Float64(y - x)) / t));
	else
		tmp = fma(Float64(y - x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.25e+54], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -2.65e-130], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.1e-19], N[(y - N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.24999999999999992e54

   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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 93.0

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 72.7%

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

   if -2.24999999999999992e54 < a < -2.6500000000000002e-130

   1. Initial program 70.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 81.9

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

   3. Applied rewrites 81.9%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 76.7

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 60.4

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

   9. Applied rewrites 60.4%

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

   if -2.6500000000000002e-130 < a < 1.0999999999999999e-19

   1. Initial program 66.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 76.9

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 66.7

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

   6. Applied rewrites 66.7%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 77.8

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in a around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift--.f64 74.4

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

   12. Applied rewrites 74.4%

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

   if 1.0999999999999999e-19 < a

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.7

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

   3. Applied rewrites 89.7%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 65.6

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

   6. Applied rewrites 65.6%

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

Alternative 4: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.45 \cdot 10^{-169}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;a \leq 6.6:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.4e+40)
   (fma y (/ (- z t) a) x)
   (if (<= a 4.45e-169)
     (* (- y x) (/ z (- a t)))
     (if (<= a 6.6) (/ (* (- z t) y) (- a t)) (fma (- y x) (/ z a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.4e+40) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= 4.45e-169) {
		tmp = (y - x) * (z / (a - t));
	} else if (a <= 6.6) {
		tmp = ((z - t) * y) / (a - t);
	} else {
		tmp = fma((y - x), (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.4e+40)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= 4.45e-169)
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	elseif (a <= 6.6)
		tmp = Float64(Float64(Float64(z - t) * y) / Float64(a - t));
	else
		tmp = fma(Float64(y - x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.4e+40], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 4.45e-169], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.6], N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.4e40

   1. Initial program 69.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.8

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

   3. Applied rewrites 92.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 71.4%

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

   if -2.4e40 < a < 4.44999999999999971e-169

   1. Initial program 67.2%

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

   2. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 53.7

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

   4. Applied rewrites 53.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 56.6

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

   6. Applied rewrites 56.6%

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

   if 4.44999999999999971e-169 < a < 6.5999999999999996

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 49.2

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

   4. Applied rewrites 49.2%

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

   if 6.5999999999999996 < a

   1. Initial program 67.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.9

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

   3. Applied rewrites 89.9%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 66.7

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

   6. Applied rewrites 66.7%

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

Alternative 5: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + -1 \cdot \left(x \cdot \frac{a - z}{t}\right)\\ \mathbf{if}\;t \leq -3 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (* -1.0 (* x (/ (- a z) t))))))
   (if (<= t -3e+199)
     t_1
     (if (<= t 6.8e+165) (fma (- y x) (/ (- z t) (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y + (-1.0 * (x * ((a - z) / t)));
	double tmp;
	if (t <= -3e+199) {
		tmp = t_1;
	} else if (t <= 6.8e+165) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y + Float64(-1.0 * Float64(x * Float64(Float64(a - z) / t))))
	tmp = 0.0
	if (t <= -3e+199)
		tmp = t_1;
	elseif (t <= 6.8e+165)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y + N[(-1.0 * N[(x * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e+199], t$95$1, If[LessEqual[t, 6.8e+165], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0000000000000001e199 or 6.80000000000000022e165 < t

   1. Initial program 29.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 65.6

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

   3. Applied rewrites 65.6%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 48.8

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

   6. Applied rewrites 48.8%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 65.1

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

   9. Applied rewrites 65.1%

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

   10. Taylor expanded in x around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. sub-div N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 81.6

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

   12. Applied rewrites 81.6%

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

   if -3.0000000000000001e199 < t < 6.80000000000000022e165

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.1

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

   3. Applied rewrites 89.1%

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

Alternative 6: 58.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+196}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.6e+196)
   y
   (if (<= t -1.7e-188)
     (fma y (/ (- z t) a) x)
     (if (<= t 4e+116) (fma z (/ (- y x) a) x) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.6e+196) {
		tmp = y;
	} else if (t <= -1.7e-188) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t <= 4e+116) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.6e+196)
		tmp = y;
	elseif (t <= -1.7e-188)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t <= 4e+116)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.6e+196], y, If[LessEqual[t, -1.7e-188], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 4e+116], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.59999999999999996e196 or 4.00000000000000006e116 < t

   1. Initial program 32.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 55.9%

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

   if -1.59999999999999996e196 < t < -1.70000000000000014e-188

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 85.7

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 46.3%

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

   if -1.70000000000000014e-188 < t < 4.00000000000000006e116

   1. Initial program 86.3%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 67.6

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

   4. Applied rewrites 67.6%

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

Alternative 7: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \mathbf{if}\;a \leq -3.9 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;y + z \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z t) (- a t)) x)))
   (if (<= a -3.9e-126)
     t_1
     (if (<= a 5.6e-59) (+ y (* z (/ (- x y) t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((z - t) / (a - t)), x);
	double tmp;
	if (a <= -3.9e-126) {
		tmp = t_1;
	} else if (a <= 5.6e-59) {
		tmp = y + (z * ((x - y) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - t) / Float64(a - t)), x)
	tmp = 0.0
	if (a <= -3.9e-126)
		tmp = t_1;
	elseif (a <= 5.6e-59)
		tmp = Float64(y + Float64(z * Float64(Float64(x - y) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.9e-126], t$95$1, If[LessEqual[a, 5.6e-59], N[(y + N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8999999999999998e-126 or 5.59999999999999961e-59 < a

   1. Initial program 69.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.3%

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

   if -3.8999999999999998e-126 < a < 5.59999999999999961e-59

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 76.4

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 65.8

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

   6. Applied rewrites 65.8%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 79.4

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

   9. Applied rewrites 79.4%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sub-div N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 79.3

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

   12. Applied rewrites 79.3%

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

Alternative 8: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{if}\;a \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;y + z \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) a) x)))
   (if (<= a -500000000000.0)
     t_1
     (if (<= a 4.2e-10) (+ y (* z (/ (- x y) t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / a), x);
	double tmp;
	if (a <= -500000000000.0) {
		tmp = t_1;
	} else if (a <= 4.2e-10) {
		tmp = y + (z * ((x - y) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / a), x)
	tmp = 0.0
	if (a <= -500000000000.0)
		tmp = t_1;
	elseif (a <= 4.2e-10)
		tmp = Float64(y + Float64(z * Float64(Float64(x - y) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -500000000000.0], t$95$1, If[LessEqual[a, 4.2e-10], N[(y + N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5e11 or 4.2e-10 < a

   1. Initial program 68.5%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 74.6

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

   4. Applied rewrites 74.6%

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

   if -5e11 < a < 4.2e-10

   1. Initial program 67.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 77.8

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 73.9

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

   9. Applied rewrites 73.9%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sub-div N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 73.6

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

   12. Applied rewrites 73.6%

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

Alternative 9: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;y + z \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.5e+80)
   (fma y (/ (- z t) a) x)
   (if (<= a 4.2e-10) (+ y (* z (/ (- x y) t))) (fma (- y x) (/ z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.5e+80) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= 4.2e-10) {
		tmp = y + (z * ((x - y) / t));
	} else {
		tmp = fma((y - x), (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.5e+80)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= 4.2e-10)
		tmp = Float64(y + Float64(z * Float64(Float64(x - y) / t)));
	else
		tmp = fma(Float64(y - x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.5e+80], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 4.2e-10], N[(y + N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4999999999999998e80

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 93.1

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

   3. Applied rewrites 93.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 73.9%

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

   if -2.4999999999999998e80 < a < 4.2e-10

   1. Initial program 68.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 78.9

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

   3. Applied rewrites 78.9%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 70.4

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 70.2

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

   9. Applied rewrites 70.2%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sub-div N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 70.2

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

   12. Applied rewrites 70.2%

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

   if 4.2e-10 < a

   1. Initial program 68.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.9

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

   3. Applied rewrites 89.9%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 66.4

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

   6. Applied rewrites 66.4%

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

Alternative 10: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-19}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -8.5e+40)
   (fma y (/ (- z t) a) x)
   (if (<= a 1.1e-19) (- y (/ (* z (- y x)) t)) (fma (- y x) (/ z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -8.5e+40) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= 1.1e-19) {
		tmp = y - ((z * (y - x)) / t);
	} else {
		tmp = fma((y - x), (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -8.5e+40)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= 1.1e-19)
		tmp = Float64(y - Float64(Float64(z * Float64(y - x)) / t));
	else
		tmp = fma(Float64(y - x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -8.5e+40], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.1e-19], N[(y - N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.49999999999999996e40

   1. Initial program 69.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.8

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

   3. Applied rewrites 92.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 71.4%

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

   if -8.49999999999999996e40 < a < 1.0999999999999999e-19

   1. Initial program 67.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 78.1

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

   3. Applied rewrites 78.1%

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

   4. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 69.1

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

   6. Applied rewrites 69.1%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   8. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 72.7

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

   9. Applied rewrites 72.7%

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

   10. Taylor expanded in a around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift--.f64 69.0

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

   12. Applied rewrites 69.0%

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

   if 1.0999999999999999e-19 < a

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 89.7

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

   3. Applied rewrites 89.7%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 65.6

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

   6. Applied rewrites 65.6%

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

Alternative 11: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-10}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.4e+40)
   (fma y (/ (- z t) a) x)
   (if (<= a 4.1e-10) (* (- y x) (/ z (- a t))) (fma (- y x) (/ z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.4e+40) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= 4.1e-10) {
		tmp = (y - x) * (z / (a - t));
	} else {
		tmp = fma((y - x), (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.4e+40)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= 4.1e-10)
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	else
		tmp = fma(Float64(y - x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.4e+40], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 4.1e-10], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4e40

   1. Initial program 69.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.8

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

   3. Applied rewrites 92.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 71.4%

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

   if -2.4e40 < a < 4.0999999999999998e-10

   1. Initial program 67.7%

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

   2. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{z}{a - t}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{z}{a - t}} \]

      7. lift--.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \frac{\color{blue}{z}}{a - t} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \frac{z}{\color{blue}{a - t}} \]

      9. lift--.f64 55.5

         \[\leadsto \left(y - x\right) \cdot \frac{z}{a - \color{blue}{t}} \]

   6. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} \]

   if 4.0999999999999998e-10 < a

   1. Initial program 68.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. lift--.f64 N/A

         \[\leadsto x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      3. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right)} \cdot \left(z - t\right)}{a - t} \]

      5. lift--.f64 N/A

         \[\leadsto x + \frac{\left(y - x\right) \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      9. sub-div N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\frac{z}{a - t} - \frac{t}{a - t}\right)} + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t} - \frac{t}{a - t}, x\right) \]

      12. sub-div N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a - t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a - t}}, x\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a - t}, x\right) \]

      15. lift--.f64 89.9

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z - t}{\color{blue}{a - t}}, x\right) \]

   3. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a}}, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 66.4

         \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a}}, x\right) \]

   6. Applied rewrites 66.4%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a}}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+66}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.5e+66) y (if (<= t 1.8e+117) (fma (- y x) (/ z a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.5e+66) {
		tmp = y;
	} else if (t <= 1.8e+117) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.5e+66)
		tmp = y;
	elseif (t <= 1.8e+117)
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.5e+66], y, If[LessEqual[t, 1.8e+117], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -2.49999999999999996e66 or 1.80000000000000006e117 < t

   1. Initial program 37.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{y} \]

   if -2.49999999999999996e66 < t < 1.80000000000000006e117

   1. Initial program 85.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. lift--.f64 N/A

         \[\leadsto x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      3. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right)} \cdot \left(z - t\right)}{a - t} \]

      5. lift--.f64 N/A

         \[\leadsto x + \frac{\left(y - x\right) \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      9. sub-div N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\frac{z}{a - t} - \frac{t}{a - t}\right)} + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t} - \frac{t}{a - t}, x\right) \]

      12. sub-div N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a - t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a - t}}, x\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a - t}, x\right) \]

      15. lift--.f64 92.3

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z - t}{\color{blue}{a - t}}, x\right) \]

   3. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a}}, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 66.1

         \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a}}, x\right) \]

   6. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+66}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.5e+66) y (if (<= t 4e+116) (fma z (/ (- y x) a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.5e+66) {
		tmp = y;
	} else if (t <= 4e+116) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.5e+66)
		tmp = y;
	elseif (t <= 4e+116)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.5e+66], y, If[LessEqual[t, 4e+116], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -2.49999999999999996e66 or 4.00000000000000006e116 < t

   1. Initial program 37.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{y} \]

   if -2.49999999999999996e66 < t < 4.00000000000000006e116

   1. Initial program 85.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{z \cdot \left(y - x\right)}{a} + \color{blue}{x} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y - x}{a}}, x\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{y - x}{\color{blue}{a}}, x\right) \]

      5. lift--.f64 64.5

         \[\leadsto \mathsf{fma}\left(z, \frac{y - x}{a}, x\right) \]

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+38}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.8e+38) y (if (<= t 1.18e+117) (fma y (/ z a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.8e+38) {
		tmp = y;
	} else if (t <= 1.18e+117) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.8e+38)
		tmp = y;
	elseif (t <= 1.18e+117)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.8e+38], y, If[LessEqual[t, 1.18e+117], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -4.80000000000000035e38 or 1.18e117 < t

   1. Initial program 39.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.0%

      \[\leadsto \color{blue}{y} \]

   if -4.80000000000000035e38 < t < 1.18e117

   1. Initial program 85.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. lift--.f64 N/A

         \[\leadsto x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      3. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right)} \cdot \left(z - t\right)}{a - t} \]

      5. lift--.f64 N/A

         \[\leadsto x + \frac{\left(y - x\right) \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      9. sub-div N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\frac{z}{a - t} - \frac{t}{a - t}\right)} + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t} - \frac{t}{a - t}, x\right) \]

      12. sub-div N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a - t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a - t}}, x\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a - t}, x\right) \]

      15. lift--.f64 92.5

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z - t}{\color{blue}{a - t}}, x\right) \]

   3. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{y}, \frac{z - t}{a - t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 72.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y}, \frac{z - t}{a - t}, x\right) \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{z - t}{\color{blue}{a}}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 60.0%

      \[\leadsto \mathsf{fma}\left(y, \frac{z - t}{\color{blue}{a}}, x\right) \]

   10. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{z}}{a}, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 55.7%

       \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{z}}{a}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.7 \cdot 10^{+53}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.7e+53) x (if (<= a 2.45e-8) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.7e+53) {
		tmp = x;
	} else if (a <= 2.45e-8) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-6.7d+53)) then
        tmp = x
    else if (a <= 2.45d-8) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.7e+53) {
		tmp = x;
	} else if (a <= 2.45e-8) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -6.7e+53:
		tmp = x
	elif a <= 2.45e-8:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -6.7e+53)
		tmp = x;
	elseif (a <= 2.45e-8)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -6.7e+53)
		tmp = x;
	elseif (a <= 2.45e-8)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -6.7e+53], x, If[LessEqual[a, 2.45e-8], y, x]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6999999999999997e53 or 2.4500000000000001e-8 < a

   1. Initial program 68.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.0%

      \[\leadsto \color{blue}{x} \]

   if -6.6999999999999997e53 < a < 2.4500000000000001e-8

   1. Initial program 68.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 33.9%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 24.9% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x;
}

Python

def code(x, y, z, t, a):
	return x

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 68.1%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 24.9%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Developer Target 1: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025105 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))