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

Percentage Accurate: 68.3% → 84.0%

Time: 8.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{x - t}{\frac{a - z}{z - y}}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-32}:\\ \;\;\;\;t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4e-11)
   (+ x (/ (- x t) (/ (- a z) (- z y))))
   (if (<= a 9e-32)
     (+ t (* -1.0 (/ (- (* y (- t x)) (* a (- t x))) z)))
     (fma (/ (- x t) (- z a)) (- y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4e-11) {
		tmp = x + ((x - t) / ((a - z) / (z - y)));
	} else if (a <= 9e-32) {
		tmp = t + (-1.0 * (((y * (t - x)) - (a * (t - x))) / z));
	} else {
		tmp = fma(((x - t) / (z - a)), (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4e-11)
		tmp = Float64(x + Float64(Float64(x - t) / Float64(Float64(a - z) / Float64(z - y))));
	elseif (a <= 9e-32)
		tmp = Float64(t + Float64(-1.0 * Float64(Float64(Float64(y * Float64(t - x)) - Float64(a * Float64(t - x))) / z)));
	else
		tmp = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4e-11], N[(x + N[(N[(x - t), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-32], N[(t + N[(-1.0 * N[(N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.99999999999999976e-11

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 79.1

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

   3. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. div-flip N/A

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

      22. mult-flip-rev N/A

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

   5. Applied rewrites 83.4%

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

   if -3.99999999999999976e-11 < a < 9.00000000000000009e-32

   1. Initial program 68.3%

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

   2. Taylor expanded in z around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.3

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

   4. Applied rewrites 46.3%

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

   if 9.00000000000000009e-32 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 79.2

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

   3. Applied rewrites 79.2%

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

Alternative 2: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+166}:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x - t}{\frac{a - z}{z - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e+166)
   (* t (- (/ y (- a z)) (/ z (- a z))))
   (+ x (/ (- x t) (/ (- a z) (- z y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+166) {
		tmp = t * ((y / (a - z)) - (z / (a - z)));
	} else {
		tmp = x + ((x - t) / ((a - z) / (z - y)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d+166)) then
        tmp = t * ((y / (a - z)) - (z / (a - z)))
    else
        tmp = x + ((x - t) / ((a - z) / (z - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+166) {
		tmp = t * ((y / (a - z)) - (z / (a - z)));
	} else {
		tmp = x + ((x - t) / ((a - z) / (z - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e+166:
		tmp = t * ((y / (a - z)) - (z / (a - z)))
	else:
		tmp = x + ((x - t) / ((a - z) / (z - y)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e+166)
		tmp = Float64(t * Float64(Float64(y / Float64(a - z)) - Float64(z / Float64(a - z))));
	else
		tmp = Float64(x + Float64(Float64(x - t) / Float64(Float64(a - z) / Float64(z - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e+166)
		tmp = t * ((y / (a - z)) - (z / (a - z)));
	else
		tmp = x + ((x - t) / ((a - z) / (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e+166], N[(t * N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x - t), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999976e166

   1. Initial program 68.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 51.4

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

   4. Applied rewrites 51.4%

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

   if -3.99999999999999976e166 < z

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 79.1

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

   3. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. div-flip N/A

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

      22. mult-flip-rev N/A

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

   5. Applied rewrites 83.4%

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

Alternative 3: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x - t}{\frac{a - z}{z - y}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + ((x - t) / ((a - z) / (z - 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 = x + ((x - t) / ((a - z) / (z - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. lift--.f64 N/A

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

   8. sub-negate-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lower--.f64 79.1

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

3. Applied rewrites 79.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lift--.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. associate-*l/ N/A

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

   11. associate-/l* N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lift--.f64 N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lift--.f64 N/A

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

   18. frac-2neg N/A

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

   19. lift-/.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. div-flip N/A

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

   22. mult-flip-rev N/A

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

5. Applied rewrites 83.4%

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

Alternative 4: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. frac-2neg N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. lower--.f64 83.3

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

3. Applied rewrites 83.3%

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

Alternative 5: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a - z}, y - z, x\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z - a}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t (- a z)) (- y z) x)))
   (if (<= t -4.5e-60)
     t_1
     (if (<= t 6.6e-140) (fma (/ x (- z a)) (- y z) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / (a - z)), (y - z), x);
	double tmp;
	if (t <= -4.5e-60) {
		tmp = t_1;
	} else if (t <= 6.6e-140) {
		tmp = fma((x / (z - a)), (y - z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / Float64(a - z)), Float64(y - z), x)
	tmp = 0.0
	if (t <= -4.5e-60)
		tmp = t_1;
	elseif (t <= 6.6e-140)
		tmp = fma(Float64(x / Float64(z - a)), Float64(y - z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -4.5e-60], t$95$1, If[LessEqual[t, 6.6e-140], N[(N[(x / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.50000000000000001e-60 or 6.59999999999999975e-140 < t

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 63.4

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

   6. Applied rewrites 63.4%

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

   if -4.50000000000000001e-60 < t < 6.59999999999999975e-140

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 79.2

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 41.1

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

   6. Applied rewrites 41.1%

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

Alternative 6: 58.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9200000000:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(t - x, 1 - \frac{y}{z}, x\right)\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9200000000.0)
   (fma (- t x) (/ z (- z a)) x)
   (if (<= a -8.6e-96)
     (fma (- t x) (- 1.0 (/ y z)) x)
     (if (<= a 7.6e-16) (/ (* t (- y z)) (- a z)) (fma (- t x) (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9200000000.0) {
		tmp = fma((t - x), (z / (z - a)), x);
	} else if (a <= -8.6e-96) {
		tmp = fma((t - x), (1.0 - (y / z)), x);
	} else if (a <= 7.6e-16) {
		tmp = (t * (y - z)) / (a - z);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9200000000.0)
		tmp = fma(Float64(t - x), Float64(z / Float64(z - a)), x);
	elseif (a <= -8.6e-96)
		tmp = fma(Float64(t - x), Float64(1.0 - Float64(y / z)), x);
	elseif (a <= 7.6e-16)
		tmp = Float64(Float64(t * Float64(y - z)) / Float64(a - z));
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9200000000.0], N[(N[(t - x), $MachinePrecision] * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -8.6e-96], N[(N[(t - x), $MachinePrecision] * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 7.6e-16], N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -9.2e9

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 46.1%

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

   if -9.2e9 < a < -8.59999999999999961e-96

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 38.0

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

   6. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-flip N/A

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

      5. *-inverses N/A

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

      6. sub-flip N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 38.0

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

   8. Applied rewrites 38.0%

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

   if -8.59999999999999961e-96 < a < 7.60000000000000024e-16

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.3

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

   4. Applied rewrites 39.3%

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

   if 7.60000000000000024e-16 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 48.8

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

   6. Applied rewrites 48.8%

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

Alternative 7: 58.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -2500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(t - x, 1 - \frac{y}{z}, x\right)\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -2500000000.0)
     t_1
     (if (<= a -8.6e-96)
       (fma (- t x) (- 1.0 (/ y z)) x)
       (if (<= a 7.6e-16) (/ (* t (- y z)) (- a z)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (y / a), x);
	double tmp;
	if (a <= -2500000000.0) {
		tmp = t_1;
	} else if (a <= -8.6e-96) {
		tmp = fma((t - x), (1.0 - (y / z)), x);
	} else if (a <= 7.6e-16) {
		tmp = (t * (y - z)) / (a - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -2500000000.0)
		tmp = t_1;
	elseif (a <= -8.6e-96)
		tmp = fma(Float64(t - x), Float64(1.0 - Float64(y / z)), x);
	elseif (a <= 7.6e-16)
		tmp = Float64(Float64(t * Float64(y - z)) / Float64(a - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2500000000.0], t$95$1, If[LessEqual[a, -8.6e-96], N[(N[(t - x), $MachinePrecision] * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 7.6e-16], N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.5e9 or 7.60000000000000024e-16 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 48.8

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

   6. Applied rewrites 48.8%

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

   if -2.5e9 < a < -8.59999999999999961e-96

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 38.0

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

   6. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-flip N/A

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

      5. *-inverses N/A

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

      6. sub-flip N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 38.0

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

   8. Applied rewrites 38.0%

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

   if -8.59999999999999961e-96 < a < 7.60000000000000024e-16

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.3

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

   4. Applied rewrites 39.3%

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

Alternative 8: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -3000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -3000000000.0)
     t_1
     (if (<= a 7.6e-16) (/ (* t (- y z)) (- a z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (y / a), x);
	double tmp;
	if (a <= -3000000000.0) {
		tmp = t_1;
	} else if (a <= 7.6e-16) {
		tmp = (t * (y - z)) / (a - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -3000000000.0)
		tmp = t_1;
	elseif (a <= 7.6e-16)
		tmp = Float64(Float64(t * Float64(y - z)) / Float64(a - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3000000000.0], t$95$1, If[LessEqual[a, 7.6e-16], N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3e9 or 7.60000000000000024e-16 < a

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 48.8

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

   6. Applied rewrites 48.8%

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

   if -3e9 < a < 7.60000000000000024e-16

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.3

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

   4. Applied rewrites 39.3%

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

Alternative 9: 56.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+36}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.8e+36)
   (+ x t)
   (if (<= z 2.3e+33) (fma (- t x) (/ y a) x) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.8e+36) {
		tmp = x + t;
	} else if (z <= 2.3e+33) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = x + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.8e+36)
		tmp = Float64(x + t);
	elseif (z <= 2.3e+33)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.8e+36], N[(x + t), $MachinePrecision], If[LessEqual[z, 2.3e+33], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.8e36 or 2.30000000000000011e33 < z

   1. Initial program 68.3%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.1

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto x + -1 \cdot x \]

   7. Applied rewrites 2.8%

      \[\leadsto x + -1 \cdot \color{blue}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto x + t \]
   9. Step-by-step derivation

   10. Applied rewrites 33.9%

       \[\leadsto x + t \]

   if -5.8e36 < z < 2.30000000000000011e33

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 48.8

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

   6. Applied rewrites 48.8%

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

Alternative 10: 51.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+55}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{z - a} \cdot x\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.15e+55)
   (+ x t)
   (if (<= z -1.9e+14)
     (* (/ y (- z a)) x)
     (if (<= z 1.82e+21) (fma t (/ (- y z) a) x) (+ x t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.15e+55) {
		tmp = x + t;
	} else if (z <= -1.9e+14) {
		tmp = (y / (z - a)) * x;
	} else if (z <= 1.82e+21) {
		tmp = fma(t, ((y - z) / a), x);
	} else {
		tmp = x + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.15e+55)
		tmp = Float64(x + t);
	elseif (z <= -1.9e+14)
		tmp = Float64(Float64(y / Float64(z - a)) * x);
	elseif (z <= 1.82e+21)
		tmp = fma(t, Float64(Float64(y - z) / a), x);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.15e+55], N[(x + t), $MachinePrecision], If[LessEqual[z, -1.9e+14], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.82e+21], N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999994e55 or 1.82e21 < z

   1. Initial program 68.3%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.1

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto x + -1 \cdot x \]

   7. Applied rewrites 2.8%

      \[\leadsto x + -1 \cdot \color{blue}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto x + t \]
   9. Step-by-step derivation

   10. Applied rewrites 33.9%

       \[\leadsto x + t \]

   if -1.14999999999999994e55 < z < -1.9e14

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. div-flip-rev N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. sum-to-mult N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 24.5

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

   11. Applied rewrites 24.5%

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

   if -1.9e14 < z < 1.82e21

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 45.3

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

   9. Applied rewrites 45.3%

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

Alternative 11: 42.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z - a} \cdot x\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{+136}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- z a)) x)))
   (if (<= y -1.1e+63) t_1 (if (<= y 3.25e+136) (+ x t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * x;
	double tmp;
	if (y <= -1.1e+63) {
		tmp = t_1;
	} else if (y <= 3.25e+136) {
		tmp = x + t;
	} 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 / (z - a)) * x
    if (y <= (-1.1d+63)) then
        tmp = t_1
    else if (y <= 3.25d+136) then
        tmp = x + t
    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 / (z - a)) * x;
	double tmp;
	if (y <= -1.1e+63) {
		tmp = t_1;
	} else if (y <= 3.25e+136) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (z - a)) * x
	tmp = 0
	if y <= -1.1e+63:
		tmp = t_1
	elif y <= 3.25e+136:
		tmp = x + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(z - a)) * x)
	tmp = 0.0
	if (y <= -1.1e+63)
		tmp = t_1;
	elseif (y <= 3.25e+136)
		tmp = Float64(x + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (z - a)) * x;
	tmp = 0.0;
	if (y <= -1.1e+63)
		tmp = t_1;
	elseif (y <= 3.25e+136)
		tmp = x + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -1.1e+63], t$95$1, If[LessEqual[y, 3.25e+136], N[(x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0999999999999999e63 or 3.2499999999999999e136 < y

   1. Initial program 68.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 83.3

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

   3. Applied rewrites 83.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. div-flip-rev N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. sum-to-mult N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 24.5

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

   11. Applied rewrites 24.5%

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

   if -1.0999999999999999e63 < y < 3.2499999999999999e136

   1. Initial program 68.3%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.1

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto x + -1 \cdot x \]

   7. Applied rewrites 2.8%

      \[\leadsto x + -1 \cdot \color{blue}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto x + t \]
   9. Step-by-step derivation

   10. Applied rewrites 33.9%

       \[\leadsto x + t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + t), $MachinePrecision]

Derivation

1. Initial program 68.3%

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

2. Taylor expanded in z around inf

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

   1. lower--.f64 19.1

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

4. Applied rewrites 19.1%

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

5. Taylor expanded in x around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
6. Step-by-step derivation

   1. lower-*.f64 2.8

      \[\leadsto x + -1 \cdot x \]

7. Applied rewrites 2.8%

   \[\leadsto x + -1 \cdot \color{blue}{x} \]

8. Taylor expanded in x around 0

   \[\leadsto x + t \]
9. Step-by-step derivation

10. Applied rewrites 33.9%

    \[\leadsto x + t \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y z) (- t x)) (- a z))))