Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B

Percentage Accurate: 85.5% → 98.1%

Time: 3.6s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. sub-div N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-div N/A

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

   12. lower-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift--.f64 98.1

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

3. Applied rewrites 98.1%

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

Alternative 2: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{y \cdot z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (fma (- y) (/ z t) y))))
   (if (<= t -7.4e+174)
     t_1
     (if (<= t -7e-163)
       (fma (/ z (- a t)) y x)
       (if (<= t 5.2e+21) (+ x (/ (* y z) (- a t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + fma(-y, (z / t), y);
	double tmp;
	if (t <= -7.4e+174) {
		tmp = t_1;
	} else if (t <= -7e-163) {
		tmp = fma((z / (a - t)), y, x);
	} else if (t <= 5.2e+21) {
		tmp = x + ((y * z) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + fma(Float64(-y), Float64(z / t), y))
	tmp = 0.0
	if (t <= -7.4e+174)
		tmp = t_1;
	elseif (t <= -7e-163)
		tmp = fma(Float64(z / Float64(a - t)), y, x);
	elseif (t <= 5.2e+21)
		tmp = Float64(x + Float64(Float64(y * z) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[((-y) * N[(z / t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.4e+174], t$95$1, If[LessEqual[t, -7e-163], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 5.2e+21], N[(x + N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.4000000000000004e174 or 5.2e21 < t

   1. Initial program 70.7%

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

   2. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 75.5

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-/.f64 89.6

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

   10. Applied rewrites 89.6%

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

   if -7.4000000000000004e174 < t < -7.00000000000000054e-163

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 98.8

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 77.3%

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

   if -7.00000000000000054e-163 < t < 5.2e21

   1. Initial program 95.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 87.5%

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

Alternative 3: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- a t)) y x)))
   (if (<= z -1.3e-5)
     t_1
     (if (<= z 5.2e+21) (fma (/ (- t) (- a t)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / (a - t)), y, x);
	double tmp;
	if (z <= -1.3e-5) {
		tmp = t_1;
	} else if (z <= 5.2e+21) {
		tmp = fma((-t / (a - t)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (z <= -1.3e-5)
		tmp = t_1;
	elseif (z <= 5.2e+21)
		tmp = fma(Float64(Float64(-t) / Float64(a - t)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -1.3e-5], t$95$1, If[LessEqual[z, 5.2e+21], N[(N[((-t) / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999992e-5 or 5.2e21 < z

   1. Initial program 82.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 96.8

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

   3. Applied rewrites 96.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 84.5%

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

   if -1.29999999999999992e-5 < z < 5.2e21

   1. Initial program 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.6

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

   6. Applied rewrites 89.6%

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

Alternative 4: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (fma (- y) (/ z t) y))))
   (if (<= t -7.4e+174) t_1 (if (<= t 1.2e+112) (fma (/ z (- a t)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + fma(-y, (z / t), y);
	double tmp;
	if (t <= -7.4e+174) {
		tmp = t_1;
	} else if (t <= 1.2e+112) {
		tmp = fma((z / (a - t)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + fma(Float64(-y), Float64(z / t), y))
	tmp = 0.0
	if (t <= -7.4e+174)
		tmp = t_1;
	elseif (t <= 1.2e+112)
		tmp = fma(Float64(z / Float64(a - t)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[((-y) * N[(z / t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.4e+174], t$95$1, If[LessEqual[t, 1.2e+112], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.4000000000000004e174 or 1.2e112 < t

   1. Initial program 66.0%

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

   2. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-/.f64 92.7

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

   10. Applied rewrites 92.7%

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

   if -7.4000000000000004e174 < t < 1.2e112

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 97.4

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

   3. Applied rewrites 97.4%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 82.9%

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

Alternative 5: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+175}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.8e+175)
   (+ x y)
   (if (<= t 5e+148) (fma (/ z (- a t)) y x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.8e+175) {
		tmp = x + y;
	} else if (t <= 5e+148) {
		tmp = fma((z / (a - t)), y, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.8e+175)
		tmp = Float64(x + y);
	elseif (t <= 5e+148)
		tmp = fma(Float64(z / Float64(a - t)), y, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.8e+175], N[(x + y), $MachinePrecision], If[LessEqual[t, 5e+148], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.80000000000000017e175 or 5.00000000000000024e148 < t

   1. Initial program 64.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 87.6%

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

   if -1.80000000000000017e175 < t < 5.00000000000000024e148

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 82.1%

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

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{+174}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.4e+174)
   (+ x y)
   (if (<= t 5.3e+21) (fma y (/ (- z t) a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.4e+174) {
		tmp = x + y;
	} else if (t <= 5.3e+21) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.4e+174)
		tmp = Float64(x + y);
	elseif (t <= 5.3e+21)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.4e+174], N[(x + y), $MachinePrecision], If[LessEqual[t, 5.3e+21], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.4000000000000004e174 or 5.3e21 < t

   1. Initial program 70.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 80.5%

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

   if -7.4000000000000004e174 < t < 5.3e21

   1. Initial program 93.2%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.5

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

   4. Applied rewrites 73.5%

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

Alternative 7: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+21}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.7e+114) (+ x y) (if (<= t 4.5e+21) (+ x (* z (/ y a))) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.7e+114) {
		tmp = x + y;
	} else if (t <= 4.5e+21) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.7d+114)) then
        tmp = x + y
    else if (t <= 4.5d+21) then
        tmp = x + (z * (y / a))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.7e+114) {
		tmp = x + y;
	} else if (t <= 4.5e+21) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.7e+114:
		tmp = x + y
	elif t <= 4.5e+21:
		tmp = x + (z * (y / a))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.7e+114)
		tmp = Float64(x + y);
	elseif (t <= 4.5e+21)
		tmp = Float64(x + Float64(z * Float64(y / a)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.7e+114)
		tmp = x + y;
	elseif (t <= 4.5e+21)
		tmp = x + (z * (y / a));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.7e+114], N[(x + y), $MachinePrecision], If[LessEqual[t, 4.5e+21], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e114 or 4.5e21 < t

   1. Initial program 71.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 79.7%

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

   if -1.7e114 < t < 4.5e21

   1. Initial program 94.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.4

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

   4. Applied rewrites 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.3

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

   6. Applied rewrites 73.3%

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

Alternative 8: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.7e+114) (+ x y) (if (<= t 4.8e+21) (fma y (/ z a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.7e+114) {
		tmp = x + y;
	} else if (t <= 4.8e+21) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.7e+114)
		tmp = Float64(x + y);
	elseif (t <= 4.8e+21)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.7e+114], N[(x + y), $MachinePrecision], If[LessEqual[t, 4.8e+21], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e114 or 4.8e21 < t

   1. Initial program 71.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 79.7%

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

   if -1.7e114 < t < 4.8e21

   1. Initial program 94.2%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 73.5

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

   4. Applied rewrites 73.5%

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

Alternative 9: 63.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.3 \cdot 10^{-58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6.3e-58) (+ x y) (if (<= t 4.6e+21) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6.3e-58) {
		tmp = x + y;
	} else if (t <= 4.6e+21) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-6.3d-58)) then
        tmp = x + y
    else if (t <= 4.6d+21) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6.3e-58) {
		tmp = x + y;
	} else if (t <= 4.6e+21) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -6.3e-58:
		tmp = x + y
	elif t <= 4.6e+21:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -6.3e-58)
		tmp = Float64(x + y);
	elseif (t <= 4.6e+21)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -6.3e-58)
		tmp = x + y;
	elseif (t <= 4.6e+21)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -6.3e-58], N[(x + y), $MachinePrecision], If[LessEqual[t, 4.6e+21], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.29999999999999999e-58 or 4.6e21 < t

   1. Initial program 76.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 73.8%

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

   if -6.29999999999999999e-58 < t < 4.6e21

   1. Initial program 95.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.6%

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

Alternative 10: 50.8% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 50.8%

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

Developer Target 1: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- a t) (- z t)))))

  (+ x (/ (* y (- z t)) (- a t))))