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

Percentage Accurate: 98.1% → 98.1%

Time: 7.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

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(y * Float64(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[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

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(y * Float64(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[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

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(y * Float64(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[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

   \[x + y \cdot \frac{z - t}{z - a} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 0.999998:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{z - a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+22}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ (- t) z) y x)))
   (if (<= t_1 -1e+84)
     t_2
     (if (<= t_1 0.0)
       (fma (/ y a) t x)
       (if (<= t_1 0.999998)
         (fma z (/ y (- z a)) x)
         (if (<= t_1 2e+22) (+ y x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((-t / z), y, x);
	double tmp;
	if (t_1 <= -1e+84) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 0.999998) {
		tmp = fma(z, (y / (z - a)), x);
	} else if (t_1 <= 2e+22) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(Float64(-t) / z), y, x)
	tmp = 0.0
	if (t_1 <= -1e+84)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 0.999998)
		tmp = fma(z, Float64(y / Float64(z - a)), x);
	elseif (t_1 <= 2e+22)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+84], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 0.999998], N[(z * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+22], N[(y + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000006e84 or 2e22 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.8%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.3%

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

   if -1.00000000000000006e84 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0

   1. Initial program 97.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if 0.0 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.999998000000000054

   1. Initial program 99.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 84.6%

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

   if 0.999998000000000054 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e22

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

Alternative 3: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (- z t) (/ y (- z a)))))
   (if (<= t_1 -2e+58)
     t_2
     (if (<= t_1 4e-28)
       (- x (/ (* (- z t) y) a))
       (if (<= t_1 5e+28) (fma (/ z (- z a)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (z - t) * (y / (z - a));
	double tmp;
	if (t_1 <= -2e+58) {
		tmp = t_2;
	} else if (t_1 <= 4e-28) {
		tmp = x - (((z - t) * y) / a);
	} else if (t_1 <= 5e+28) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(z - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -2e+58)
		tmp = t_2;
	elseif (t_1 <= 4e-28)
		tmp = Float64(x - Float64(Float64(Float64(z - t) * y) / a));
	elseif (t_1 <= 5e+28)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+58], t$95$2, If[LessEqual[t$95$1, 4e-28], N[(x - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+28], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e58 or 4.99999999999999957e28 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -1.99999999999999989e58 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.99999999999999988e-28

   1. Initial program 98.3%

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

   3. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 3.99999999999999988e-28 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999957e28

   1. Initial program 99.9%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

Alternative 4: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (- z t) (/ y (- z a)))))
   (if (<= t_1 -2e+58)
     t_2
     (if (<= t_1 0.0)
       (fma (/ y a) t x)
       (if (<= t_1 5e+28) (fma (/ z (- z a)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (z - t) * (y / (z - a));
	double tmp;
	if (t_1 <= -2e+58) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 5e+28) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(z - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -2e+58)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 5e+28)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+58], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+28], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e58 or 4.99999999999999957e28 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -1.99999999999999989e58 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0

   1. Initial program 97.5%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 0.0 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999957e28

   1. Initial program 99.9%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 5: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+84)
     (fma (/ (- t) z) y x)
     (if (<= t_1 0.0)
       (fma (/ y a) t x)
       (if (<= t_1 5e+28) (fma (/ z (- z a)) y x) (* (- y) (/ t (- z a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -1e+84) {
		tmp = fma((-t / z), y, x);
	} else if (t_1 <= 0.0) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 5e+28) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = -y * (t / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -1e+84)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 5e+28)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(-y) * Float64(t / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+84], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+28], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[((-y) * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000006e84

   1. Initial program 87.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.3%

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

   if -1.00000000000000006e84 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0

   1. Initial program 97.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if 0.0 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999957e28

   1. Initial program 99.9%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 4.99999999999999957e28 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 86.4

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

   5. Applied rewrites 86.4%

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

Alternative 6: 81.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq 5000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+84)
     (fma (/ (- t) z) y x)
     (if (or (<= t_1 0.0) (not (<= t_1 5000.0)))
       (fma (/ y a) t x)
       (fma (/ z (- z a)) y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -1e+84) {
		tmp = fma((-t / z), y, x);
	} else if ((t_1 <= 0.0) || !(t_1 <= 5000.0)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = fma((z / (z - a)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -1e+84)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif ((t_1 <= 0.0) || !(t_1 <= 5000.0))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+84], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 5000.0]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000006e84

   1. Initial program 87.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.3%

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

   if -1.00000000000000006e84 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0 or 5e3 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.4%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 0.0 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e3

   1. Initial program 99.9%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 94.2

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

   5. Applied rewrites 94.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0 \lor \neg \left(\frac{z - t}{z - a} \leq 5000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+22}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ (- t) z) y x)))
   (if (<= t_1 -1e+84)
     t_2
     (if (<= t_1 0.001) (fma (/ y a) t x) (if (<= t_1 2e+22) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((-t / z), y, x);
	double tmp;
	if (t_1 <= -1e+84) {
		tmp = t_2;
	} else if (t_1 <= 0.001) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 2e+22) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(Float64(-t) / z), y, x)
	tmp = 0.0
	if (t_1 <= -1e+84)
		tmp = t_2;
	elseif (t_1 <= 0.001)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 2e+22)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+84], t$95$2, If[LessEqual[t$95$1, 0.001], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+22], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000006e84 or 2e22 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.8%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.3%

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

   if -1.00000000000000006e84 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-3

   1. Initial program 98.4%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if 1e-3 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e22

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.6

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

   5. Applied rewrites 94.6%

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

Alternative 8: 69.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+256}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-200}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* y t) a)))
   (if (<= t_1 -4e+256)
     t_2
     (if (<= t_1 4e-200) (* 1.0 x) (if (<= t_1 5e+28) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / a;
	double tmp;
	if (t_1 <= -4e+256) {
		tmp = t_2;
	} else if (t_1 <= 4e-200) {
		tmp = 1.0 * x;
	} else if (t_1 <= 5e+28) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	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 = (z - t) / (z - a)
    t_2 = (y * t) / a
    if (t_1 <= (-4d+256)) then
        tmp = t_2
    else if (t_1 <= 4d-200) then
        tmp = 1.0d0 * x
    else if (t_1 <= 5d+28) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / a;
	double tmp;
	if (t_1 <= -4e+256) {
		tmp = t_2;
	} else if (t_1 <= 4e-200) {
		tmp = 1.0 * x;
	} else if (t_1 <= 5e+28) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (y * t) / a
	tmp = 0
	if t_1 <= -4e+256:
		tmp = t_2
	elif t_1 <= 4e-200:
		tmp = 1.0 * x
	elif t_1 <= 5e+28:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y * t) / a)
	tmp = 0.0
	if (t_1 <= -4e+256)
		tmp = t_2;
	elseif (t_1 <= 4e-200)
		tmp = Float64(1.0 * x);
	elseif (t_1 <= 5e+28)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (y * t) / a;
	tmp = 0.0;
	if (t_1 <= -4e+256)
		tmp = t_2;
	elseif (t_1 <= 4e-200)
		tmp = 1.0 * x;
	elseif (t_1 <= 5e+28)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+256], t$95$2, If[LessEqual[t$95$1, 4e-200], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 5e+28], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.0000000000000001e256 or 4.99999999999999957e28 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 88.2%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.2%

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

   if -4.0000000000000001e256 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.9999999999999999e-200

   1. Initial program 98.4%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 81.4

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

   8. Applied rewrites 81.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 56.7%

       \[\leadsto 1 \cdot x \]

   if 3.9999999999999999e-200 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999957e28

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.2

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

   5. Applied rewrites 84.2%

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

Alternative 9: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.001 \lor \neg \left(t\_1 \leq 5000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 0.001) (not (<= t_1 5000.0))) (fma (/ y a) t x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 0.001) || !(t_1 <= 5000.0)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 0.001) || !(t_1 <= 5000.0))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.001], N[Not[LessEqual[t$95$1, 5000.0]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-3 or 5e3 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.0%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if 1e-3 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e3

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 0.001 \lor \neg \left(\frac{z - t}{z - a} \leq 5000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.001 \lor \neg \left(t\_1 \leq 5000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 0.001) (not (<= t_1 5000.0))) (fma y (/ t a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 0.001) || !(t_1 <= 5000.0)) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 0.001) || !(t_1 <= 5000.0))
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.001], N[Not[LessEqual[t$95$1, 5000.0]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-3 or 5e3 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.0%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 69.2%

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

   if 1e-3 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e3

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 0.001 \lor \neg \left(\frac{z - t}{z - a} \leq 5000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+29} \lor \neg \left(z \leq 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7e+29) (not (<= z 1e-35)))
   (fma (/ (- z t) z) y x)
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7e+29) || !(z <= 1e-35)) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7e+29) || !(z <= 1e-35))
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7e+29], N[Not[LessEqual[z, 1e-35]], $MachinePrecision]], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999958e29 or 1.00000000000000001e-35 < z

   1. Initial program 99.8%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -6.99999999999999958e29 < z < 1.00000000000000001e-35

   1. Initial program 95.2%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+29} \lor \neg \left(z \leq 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 9.5 \cdot 10^{-187}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 9.5e-187) (* 1.0 x) (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 9.5e-187) {
		tmp = 1.0 * x;
	} else {
		tmp = y + 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 - t) / (z - a)) <= 9.5d-187) then
        tmp = 1.0d0 * x
    else
        tmp = y + 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 - t) / (z - a)) <= 9.5e-187) {
		tmp = 1.0 * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 9.5e-187:
		tmp = 1.0 * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 9.5e-187)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 9.5e-187)
		tmp = 1.0 * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 9.5e-187], N[(1.0 * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.49999999999999936e-187

   1. Initial program 95.0%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 82.9

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

   8. Applied rewrites 82.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 51.4%

       \[\leadsto 1 \cdot x \]

   if 9.49999999999999936e-187 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.5

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

   5. Applied rewrites 70.5%

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

Alternative 13: 60.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 58.0

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

5. Applied rewrites 58.0%

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

Developer Target 1: 98.3% 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 2024363 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine 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)))))