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

Percentage Accurate: 86.0% → 98.0%

Time: 3.6s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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)) / (z - a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) / (z - a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. sub-div N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-div N/A

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

   12. lower-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift--.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 2: 60.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot t\\ t_2 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-46}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_2 \leq 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq 10^{+156}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y a) t)) (t_2 (/ (* y (- z t)) (- z a))))
   (if (<= t_2 -4e+118)
     t_1
     (if (<= t_2 -2e-46)
       (+ x y)
       (if (<= t_2 1e-16) x (if (<= t_2 1e+156) (+ x y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * t;
	double t_2 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_2 <= -4e+118) {
		tmp = t_1;
	} else if (t_2 <= -2e-46) {
		tmp = x + y;
	} else if (t_2 <= 1e-16) {
		tmp = x;
	} else if (t_2 <= 1e+156) {
		tmp = x + y;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (y / a) * t
    t_2 = (y * (z - t)) / (z - a)
    if (t_2 <= (-4d+118)) then
        tmp = t_1
    else if (t_2 <= (-2d-46)) then
        tmp = x + y
    else if (t_2 <= 1d-16) then
        tmp = x
    else if (t_2 <= 1d+156) then
        tmp = x + y
    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 / a) * t;
	double t_2 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_2 <= -4e+118) {
		tmp = t_1;
	} else if (t_2 <= -2e-46) {
		tmp = x + y;
	} else if (t_2 <= 1e-16) {
		tmp = x;
	} else if (t_2 <= 1e+156) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / a) * t
	t_2 = (y * (z - t)) / (z - a)
	tmp = 0
	if t_2 <= -4e+118:
		tmp = t_1
	elif t_2 <= -2e-46:
		tmp = x + y
	elif t_2 <= 1e-16:
		tmp = x
	elif t_2 <= 1e+156:
		tmp = x + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / a) * t)
	t_2 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_2 <= -4e+118)
		tmp = t_1;
	elseif (t_2 <= -2e-46)
		tmp = Float64(x + y);
	elseif (t_2 <= 1e-16)
		tmp = x;
	elseif (t_2 <= 1e+156)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / a) * t;
	t_2 = (y * (z - t)) / (z - a);
	tmp = 0.0;
	if (t_2 <= -4e+118)
		tmp = t_1;
	elseif (t_2 <= -2e-46)
		tmp = x + y;
	elseif (t_2 <= 1e-16)
		tmp = x;
	elseif (t_2 <= 1e+156)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+118], t$95$1, If[LessEqual[t$95$2, -2e-46], N[(x + y), $MachinePrecision], If[LessEqual[t$95$2, 1e-16], x, If[LessEqual[t$95$2, 1e+156], N[(x + y), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -3.99999999999999987e118 or 9.9999999999999998e155 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 59.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 45.5

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 28.9%

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

   9. Taylor expanded in z around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-/.f64 35.9

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

   11. Applied rewrites 35.9%

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

   if -3.99999999999999987e118 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -2.00000000000000005e-46 or 9.9999999999999998e-17 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 9.9999999999999998e155

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 58.0%

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

   if -2.00000000000000005e-46 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 9.9999999999999998e-17

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.4%

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

Alternative 3: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+223}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.8e+223)
   (+ x y)
   (if (<= z -7.2e+21)
     (fma (/ (- t) z) y x)
     (if (<= z 1.05e+70) (fma t (/ y a) x) (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e+223) {
		tmp = x + y;
	} else if (z <= -7.2e+21) {
		tmp = fma((-t / z), y, x);
	} else if (z <= 1.05e+70) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e+223)
		tmp = Float64(x + y);
	elseif (z <= -7.2e+21)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (z <= 1.05e+70)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.8e+223], N[(x + y), $MachinePrecision], If[LessEqual[z, -7.2e+21], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 1.05e+70], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000022e223 or 1.05000000000000004e70 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 85.0%

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

   if -4.80000000000000022e223 < z < -7.2e21

   1. Initial program 80.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 83.8%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -7.2e21 < z < 1.05000000000000004e70

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

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

Alternative 4: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-6} \lor \neg \left(z \leq 4.5 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{-a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -8.2e-6) (not (<= z 4.5e-12)))
   (fma y (/ (- z t) z) x)
   (fma (/ (- z t) (- a)) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -8.2e-6) || !(z <= 4.5e-12)) {
		tmp = fma(y, ((z - t) / z), x);
	} else {
		tmp = fma(((z - t) / -a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -8.2e-6) || !(z <= 4.5e-12))
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	else
		tmp = fma(Float64(Float64(z - t) / Float64(-a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -8.2e-6], N[Not[LessEqual[z, 4.5e-12]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / (-a)), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999994e-6 or 4.49999999999999981e-12 < z

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -8.1999999999999994e-6 < z < 4.49999999999999981e-12

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.9

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

   7. Applied rewrites 80.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-6} \lor \neg \left(z \leq 4.5 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{-a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-6} \lor \neg \left(z \leq 4.5 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -8.2e-6) (not (<= z 4.5e-12)))
   (fma y (/ (- z t) z) x)
   (fma t (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -8.2e-6) || !(z <= 4.5e-12)) {
		tmp = fma(y, ((z - t) / z), x);
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -8.2e-6) || !(z <= 4.5e-12))
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -8.2e-6], N[Not[LessEqual[z, 4.5e-12]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999994e-6 or 4.49999999999999981e-12 < z

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -8.1999999999999994e-6 < z < 4.49999999999999981e-12

   1. Initial program 95.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-6} \lor \neg \left(z \leq 4.5 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-56} \lor \neg \left(z \leq 58000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.15e-56) (not (<= z 58000.0)))
   (fma y (/ z (- z a)) x)
   (fma (/ t a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.15e-56) || !(z <= 58000.0)) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.15e-56) || !(z <= 58000.0))
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.15e-56], N[Not[LessEqual[z, 58000.0]], $MachinePrecision]], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000001e-56 or 58000 < z

   1. Initial program 77.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -1.15000000000000001e-56 < z < 58000

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 78.0

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

   7. Applied rewrites 78.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-56} \lor \neg \left(z \leq 58000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+22} \lor \neg \left(z \leq 1.05 \cdot 10^{+70}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1e+22) (not (<= z 1.05e+70))) (+ x y) (fma t (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1e+22) || !(z <= 1.05e+70)) {
		tmp = x + y;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1e+22) || !(z <= 1.05e+70))
		tmp = Float64(x + y);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1e+22], N[Not[LessEqual[z, 1.05e+70]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e22 or 1.05000000000000004e70 < z

   1. Initial program 73.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 79.7%

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

   if -1e22 < z < 1.05000000000000004e70

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+22} \lor \neg \left(z \leq 1.05 \cdot 10^{+70}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \lor \neg \left(z \leq 1.2 \cdot 10^{+49}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.9) (not (<= z 1.2e+49))) (+ x y) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.9) || !(z <= 1.2e+49)) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-3.9d0)) .or. (.not. (z <= 1.2d+49))) then
        tmp = x + y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.9) || !(z <= 1.2e+49)) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.9) or not (z <= 1.2e+49):
		tmp = x + y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.9) || !(z <= 1.2e+49))
		tmp = Float64(x + y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.9) || ~((z <= 1.2e+49)))
		tmp = x + y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.9], N[Not[LessEqual[z, 1.2e+49]], $MachinePrecision]], N[(x + y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.89999999999999991 or 1.2e49 < z

   1. Initial program 74.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.4%

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

   if -3.89999999999999991 < z < 1.2e49

   1. Initial program 95.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 50.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \lor \neg \left(z \leq 1.2 \cdot 10^{+49}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.0% 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 86.0%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 50.0%

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

Developer Target 1: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025086 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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