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

Percentage Accurate: 68.3% → 91.3%

Time: 6.0s

Alternatives: 14

Speedup: 0.7×

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: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t z) (- t a)) (- y x) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -4e-304)
     t_1
     (if (<= t_2 0.0)
       (+ y (* -1.0 (/ (- (* z (- y x)) (* a (- y x))) t)))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / (t - a)), (y - x), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -4e-304) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y + (-1.0 * (((z * (y - x)) - (a * (y - x))) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -4e-304)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(-1.0 * Float64(Float64(Float64(z * Float64(y - x)) - Float64(a * Float64(y - x))) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-304], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y + N[(-1.0 * N[(N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < -3.99999999999999988e-304 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   if -3.99999999999999988e-304 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 68.3%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.7

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

   4. Applied rewrites 46.7%

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

Alternative 2: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;t \leq -5 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) (- a t)) y)))
   (if (<= t -5e+207)
     t_1
     (if (<= t 1.5e+250) (fma (/ (- t z) (- t a)) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t <= -5e+207) {
		tmp = t_1;
	} else if (t <= 1.5e+250) {
		tmp = fma(((t - z) / (t - a)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (t <= -5e+207)
		tmp = t_1;
	elseif (t <= 1.5e+250)
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t, -5e+207], t$95$1, If[LessEqual[t, 1.5e+250], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.9999999999999999e207 or 1.49999999999999988e250 < t

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.0

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

      19. lift--.f64 N/A

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

      20. sub-negate-rev N/A

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

      21. lift--.f64 N/A

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

      22. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

   if -4.9999999999999999e207 < t < 1.49999999999999988e250

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

Alternative 3: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{elif}\;a \leq 0.00046:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.4e+95)
   (fma (/ z a) (- y x) x)
   (if (<= a 0.00046)
     (* (/ (- z t) (- a t)) y)
     (fma (/ (- t z) (- t a)) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.4e+95) {
		tmp = fma((z / a), (y - x), x);
	} else if (a <= 0.00046) {
		tmp = ((z - t) / (a - t)) * y;
	} else {
		tmp = fma(((t - z) / (t - a)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.4e+95)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	elseif (a <= 0.00046)
		tmp = Float64(Float64(Float64(z - t) / Float64(a - t)) * y);
	else
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.4e+95], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 0.00046], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999e95

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -1.3999999999999999e95 < a < 4.6000000000000001e-4

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.0

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

      19. lift--.f64 N/A

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

      20. sub-negate-rev N/A

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

      21. lift--.f64 N/A

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

      22. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

   if 4.6000000000000001e-4 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.7%

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

Alternative 4: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.014:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -1.4e+95) t_1 (if (<= a 0.014) (* (/ (- z t) (- a t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -1.4e+95) {
		tmp = t_1;
	} else if (a <= 0.014) {
		tmp = ((z - t) / (a - t)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -1.4e+95)
		tmp = t_1;
	elseif (a <= 0.014)
		tmp = Float64(Float64(Float64(z - t) / Float64(a - t)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.4e+95], t$95$1, If[LessEqual[a, 0.014], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3999999999999999e95 or 0.0140000000000000003 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -1.3999999999999999e95 < a < 0.0140000000000000003

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.0

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

      19. lift--.f64 N/A

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

      20. sub-negate-rev N/A

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

      21. lift--.f64 N/A

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

      22. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 5: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.0135:\\ \;\;\;\;\frac{y}{t - a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -1.4e+63) t_1 (if (<= a 0.0135) (* (/ y (- t a)) (- t z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -1.4e+63) {
		tmp = t_1;
	} else if (a <= 0.0135) {
		tmp = (y / (t - a)) * (t - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -1.4e+63)
		tmp = t_1;
	elseif (a <= 0.0135)
		tmp = Float64(Float64(y / Float64(t - a)) * Float64(t - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.4e+63], t$95$1, If[LessEqual[a, 0.0135], N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.39999999999999993e63 or 0.0134999999999999998 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if -1.39999999999999993e63 < a < 0.0134999999999999998

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

Alternative 6: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{t} \cdot \left(t - z\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.2e+107)
   (* (/ y t) (- t z))
   (if (<= t 7e+30) (fma (/ z a) (- y x) x) (fma (/ (- t z) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.2e+107) {
		tmp = (y / t) * (t - z);
	} else if (t <= 7e+30) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = fma(((t - z) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.2e+107)
		tmp = Float64(Float64(y / t) * Float64(t - z));
	elseif (t <= 7e+30)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.2e+107], N[(N[(y / t), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e+30], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.2e107

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 32.4

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

   9. Applied rewrites 32.4%

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

   if -2.2e107 < t < 7.00000000000000042e30

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 48.4

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

   6. Applied rewrites 48.4%

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

   if 7.00000000000000042e30 < t

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 42.4

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

   9. Applied rewrites 42.4%

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

Alternative 7: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot \left(t - z\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y t) (- t z))))
   (if (<= t -2.2e+107) t_1 (if (<= t 5e+19) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / t) * (t - z);
	double tmp;
	if (t <= -2.2e+107) {
		tmp = t_1;
	} else if (t <= 5e+19) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / t) * Float64(t - z))
	tmp = 0.0
	if (t <= -2.2e+107)
		tmp = t_1;
	elseif (t <= 5e+19)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.2e+107], t$95$1, If[LessEqual[t, 5e+19], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2e107 or 5e19 < t

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 32.4

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

   9. Applied rewrites 32.4%

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

   if -2.2e107 < t < 5e19

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 48.4

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

   6. Applied rewrites 48.4%

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

Alternative 8: 40.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot \left(t - z\right)\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{x \cdot z}{t - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y t) (- t z))))
   (if (<= y -7.4e-85) t_1 (if (<= y 5.2e-64) (/ (* x z) (- t a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / t) * (t - z);
	double tmp;
	if (y <= -7.4e-85) {
		tmp = t_1;
	} else if (y <= 5.2e-64) {
		tmp = (x * z) / (t - a);
	} 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 = (y / t) * (t - z)
    if (y <= (-7.4d-85)) then
        tmp = t_1
    else if (y <= 5.2d-64) then
        tmp = (x * z) / (t - a)
    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 = (y / t) * (t - z);
	double tmp;
	if (y <= -7.4e-85) {
		tmp = t_1;
	} else if (y <= 5.2e-64) {
		tmp = (x * z) / (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / t) * (t - z)
	tmp = 0
	if y <= -7.4e-85:
		tmp = t_1
	elif y <= 5.2e-64:
		tmp = (x * z) / (t - a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / t) * Float64(t - z))
	tmp = 0.0
	if (y <= -7.4e-85)
		tmp = t_1;
	elseif (y <= 5.2e-64)
		tmp = Float64(Float64(x * z) / Float64(t - a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / t) * (t - z);
	tmp = 0.0;
	if (y <= -7.4e-85)
		tmp = t_1;
	elseif (y <= 5.2e-64)
		tmp = (x * z) / (t - a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.4e-85], t$95$1, If[LessEqual[y, 5.2e-64], N[(N[(x * z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.39999999999999966e-85 or 5.2e-64 < y

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 32.4

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

   9. Applied rewrites 32.4%

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

   if -7.39999999999999966e-85 < y < 5.2e-64

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.0

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

   3. Applied rewrites 80.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 41.6

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 21.0

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

   9. Applied rewrites 21.0%

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

Alternative 9: 36.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot \left(t - z\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{t \cdot y}{t - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y t) (- t z))) (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+263) (/ (* t y) (- t a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / t) * (t - z);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+263) {
		tmp = (t * y) / (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / t) * (t - z);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+263) {
		tmp = (t * y) / (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / t) * (t - z)
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+263:
		tmp = (t * y) / (t - a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / t) * Float64(t - z))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+263)
		tmp = Float64(Float64(t * y) / Float64(t - a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / t) * (t - z);
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+263)
		tmp = (t * y) / (t - a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 2.00000000000000003e263 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 32.4

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

   9. Applied rewrites 32.4%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 2.00000000000000003e263

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 21.4

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

   9. Applied rewrites 21.4%

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

Alternative 10: 33.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(t - z\right)}{t}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-94}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- t z)) t)))
   (if (<= t -7e+163)
     (fma 1.0 (- y x) x)
     (if (<= t -4.7e-202) t_1 (if (<= t 1.95e-94) (* (/ z a) y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (t - z)) / t;
	double tmp;
	if (t <= -7e+163) {
		tmp = fma(1.0, (y - x), x);
	} else if (t <= -4.7e-202) {
		tmp = t_1;
	} else if (t <= 1.95e-94) {
		tmp = (z / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(t - z)) / t)
	tmp = 0.0
	if (t <= -7e+163)
		tmp = fma(1.0, Float64(y - x), x);
	elseif (t <= -4.7e-202)
		tmp = t_1;
	elseif (t <= 1.95e-94)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -7e+163], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, -4.7e-202], t$95$1, If[LessEqual[t, 1.95e-94], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.0000000000000005e163

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 20.4%

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

   if -7.0000000000000005e163 < t < -4.6999999999999999e-202 or 1.9500000000000001e-94 < t

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 27.7

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

   9. Applied rewrites 27.7%

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

   if -4.6999999999999999e-202 < t < 1.9500000000000001e-94

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.7

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 18.9

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

   9. Applied rewrites 18.9%

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

Alternative 11: 30.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{a} \cdot y\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-33}:\\ \;\;\;\;\frac{t \cdot y}{t - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ z a) y)))
   (if (<= z -1.2e+23) t_1 (if (<= z 9e-33) (/ (* t y) (- t a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z / a) * y;
	double tmp;
	if (z <= -1.2e+23) {
		tmp = t_1;
	} else if (z <= 9e-33) {
		tmp = (t * y) / (t - a);
	} 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 = (z / a) * y
    if (z <= (-1.2d+23)) then
        tmp = t_1
    else if (z <= 9d-33) then
        tmp = (t * y) / (t - a)
    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 = (z / a) * y;
	double tmp;
	if (z <= -1.2e+23) {
		tmp = t_1;
	} else if (z <= 9e-33) {
		tmp = (t * y) / (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z / a) * y
	tmp = 0
	if z <= -1.2e+23:
		tmp = t_1
	elif z <= 9e-33:
		tmp = (t * y) / (t - a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z / a) * y)
	tmp = 0.0
	if (z <= -1.2e+23)
		tmp = t_1;
	elseif (z <= 9e-33)
		tmp = Float64(Float64(t * y) / Float64(t - a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z / a) * y;
	tmp = 0.0;
	if (z <= -1.2e+23)
		tmp = t_1;
	elseif (z <= 9e-33)
		tmp = (t * y) / (t - a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.2e+23], t$95$1, If[LessEqual[z, 9e-33], N[(N[(t * y), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e23 or 8.99999999999999982e-33 < z

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.7

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 18.9

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

   9. Applied rewrites 18.9%

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

   if -1.2e23 < z < 8.99999999999999982e-33

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-rgt-neg-out N/A

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

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

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

   6. Applied rewrites 46.7%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 21.4

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

   9. Applied rewrites 21.4%

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

Alternative 12: 30.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -6.5e-93) t_1 (if (<= t 7.8e+39) (* (/ z a) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -6.5e-93) {
		tmp = t_1;
	} else if (t <= 7.8e+39) {
		tmp = (z / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -6.5e-93)
		tmp = t_1;
	elseif (t <= 7.8e+39)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -6.5e-93], t$95$1, If[LessEqual[t, 7.8e+39], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5e-93 or 7.8000000000000002e39 < t

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 84.1

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

   3. Applied rewrites 84.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 20.4%

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

   if -6.5e-93 < t < 7.8000000000000002e39

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.7

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 18.9

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

   9. Applied rewrites 18.9%

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

Alternative 13: 18.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{a} \cdot y \end{array} \]

FPCore

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

C

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

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 = (z / a) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

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. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 39.9

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

4. Applied rewrites 39.9%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 16.7

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

7. Applied rewrites 16.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 18.9

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

9. Applied rewrites 18.9%

   \[\leadsto \frac{z}{a} \cdot y \]
10. Add Preprocessing

Alternative 14: 17.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{y}{a} \cdot z \end{array} \]

FPCore

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

C

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

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 = (y / a) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

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. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 39.9

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

4. Applied rewrites 39.9%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 16.7

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

7. Applied rewrites 16.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 17.9

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

9. Applied rewrites 17.9%

   \[\leadsto \frac{y}{a} \cdot z \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y x) (- z t)) (- a t))))