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

Percentage Accurate: 76.9% → 93.6%

Time: 4.3s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\frac{t}{a - t} + 1\right) - \frac{z}{a - t}, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (fma (- (+ (/ t (- a t)) 1.0) (/ z (- a t))) y x))

C

double code(double x, double y, double z, double t, double a) {
	return fma((((t / (a - t)) + 1.0) - (z / (a - t))), y, x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(Float64(Float64(t / Float64(a - t)) + 1.0) - Float64(z / Float64(a - t))), y, x)
end

Wolfram

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

Derivation

1. Initial program 76.9%

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

2. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 93.6

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

4. Applied rewrites 93.6%

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

Alternative 2: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-1 \cdot \frac{a}{t} - \frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+101}:\\ \;\;\;\;x + \left(y - \left(z - t\right) \cdot \frac{y}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- (* -1.0 (/ a t)) (/ z (- a t))) y x)))
   (if (<= t -2.8e+99)
     t_1
     (if (<= t 6.4e+101) (+ x (- y (* (- z t) (/ y (- a t))))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((-1.0 * (a / t)) - (z / (a - t))), y, x);
	double tmp;
	if (t <= -2.8e+99) {
		tmp = t_1;
	} else if (t <= 6.4e+101) {
		tmp = x + (y - ((z - t) * (y / (a - t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-1.0 * Float64(a / t)) - Float64(z / Float64(a - t))), y, x)
	tmp = 0.0
	if (t <= -2.8e+99)
		tmp = t_1;
	elseif (t <= 6.4e+101)
		tmp = Float64(x + Float64(y - Float64(Float64(z - t) * Float64(y / Float64(a - t)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(-1.0 * N[(a / t), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.8e+99], t$95$1, If[LessEqual[t, 6.4e+101], N[(x + N[(y - N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8e99 or 6.4000000000000001e101 < t

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 65.5

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

   7. Applied rewrites 65.5%

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

   if -2.8e99 < t < 6.4000000000000001e101

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower--.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift--.f64 87.3

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

   3. Applied rewrites 87.3%

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

Alternative 3: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.7 \cdot 10^{+101}:\\ \;\;\;\;x + \left(y - \left(z - t\right) \cdot \frac{y}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (* -1.0 (/ (- a z) t)) y x)))
   (if (<= t -2.8e+99)
     t_1
     (if (<= t 7.7e+101) (+ x (- y (* (- z t) (/ y (- a t))))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-1.0 * ((a - z) / t)), y, x);
	double tmp;
	if (t <= -2.8e+99) {
		tmp = t_1;
	} else if (t <= 7.7e+101) {
		tmp = x + (y - ((z - t) * (y / (a - t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x)
	tmp = 0.0
	if (t <= -2.8e+99)
		tmp = t_1;
	elseif (t <= 7.7e+101)
		tmp = Float64(x + Float64(y - Float64(Float64(z - t) * Float64(y / Float64(a - t)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.8e+99], t$95$1, If[LessEqual[t, 7.7e+101], N[(x + N[(y - N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8e99 or 7.6999999999999996e101 < t

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 59.8

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

   7. Applied rewrites 59.8%

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

   if -2.8e99 < t < 7.6999999999999996e101

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower--.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift--.f64 87.3

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

   3. Applied rewrites 87.3%

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

Alternative 4: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-1 \cdot \frac{a - z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -4.05 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (* -1.0 (/ (- a z) t)) y x)))
   (if (<= t -4.05e+62)
     t_1
     (if (<= t 3.7e+101) (fma (- 1.0 (/ z (- a t))) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-1.0 * ((a - z) / t)), y, x);
	double tmp;
	if (t <= -4.05e+62) {
		tmp = t_1;
	} else if (t <= 3.7e+101) {
		tmp = fma((1.0 - (z / (a - t))), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-1.0 * Float64(Float64(a - z) / t)), y, x)
	tmp = 0.0
	if (t <= -4.05e+62)
		tmp = t_1;
	elseif (t <= 3.7e+101)
		tmp = fma(Float64(1.0 - Float64(z / Float64(a - t))), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(-1.0 * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -4.05e+62], t$95$1, If[LessEqual[t, 3.7e+101], N[(N[(1.0 - N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.04999999999999999e62 or 3.6999999999999997e101 < t

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 59.8

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

   7. Applied rewrites 59.8%

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

   if -4.04999999999999999e62 < t < 3.6999999999999997e101

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 81.7%

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

Alternative 5: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 3.65 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{1}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.02e+124)
   (fma (/ z t) y x)
   (if (<= t 3.65e+104)
     (fma (- 1.0 (/ z (- a t))) y x)
     (fma (* z (/ 1.0 t)) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.02e+124) {
		tmp = fma((z / t), y, x);
	} else if (t <= 3.65e+104) {
		tmp = fma((1.0 - (z / (a - t))), y, x);
	} else {
		tmp = fma((z * (1.0 / t)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.02e+124)
		tmp = fma(Float64(z / t), y, x);
	elseif (t <= 3.65e+104)
		tmp = fma(Float64(1.0 - Float64(z / Float64(a - t))), y, x);
	else
		tmp = fma(Float64(z * Float64(1.0 / t)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.02e+124], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 3.65e+104], N[(N[(1.0 - N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(z * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.01999999999999994e124

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 61.2

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

   7. Applied rewrites 61.2%

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

   if -1.01999999999999994e124 < t < 3.65000000000000025e104

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 81.7%

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

   if 3.65000000000000025e104 < t

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 86.0

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 61.2

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

   10. Applied rewrites 61.2%

       \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{t}, y, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{if}\;a \leq -1.85 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 15500000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (/ z a)) y x)))
   (if (<= a -1.85e+17) t_1 (if (<= a 15500000000.0) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((1.0 - (z / a)), y, x);
	double tmp;
	if (a <= -1.85e+17) {
		tmp = t_1;
	} else if (a <= 15500000000.0) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(1.0 - Float64(z / a)), y, x)
	tmp = 0.0
	if (a <= -1.85e+17)
		tmp = t_1;
	elseif (a <= 15500000000.0)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -1.85e+17], t$95$1, If[LessEqual[a, 15500000000.0], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.85e17 or 1.55e10 < a

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 67.5

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

   7. Applied rewrites 67.5%

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

   if -1.85e17 < a < 1.55e10

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 61.2

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

   7. Applied rewrites 61.2%

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

Alternative 7: 77.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{+23}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.12e+23) (+ x y) (if (<= a 2.5e+17) (fma (/ z t) y x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.12e+23) {
		tmp = x + y;
	} else if (a <= 2.5e+17) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.12e+23)
		tmp = Float64(x + y);
	elseif (a <= 2.5e+17)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.12e+23], N[(x + y), $MachinePrecision], If[LessEqual[a, 2.5e+17], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.12e23 or 2.5e17 < a

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 86.0

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-+.f64 60.7

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

   10. Applied rewrites 60.7%

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

   if -1.12e23 < a < 2.5e17

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 61.2

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

   7. Applied rewrites 61.2%

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

Alternative 8: 61.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 10^{+307}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (+ x y) (/ (* (- z t) y) (- a t))) 1e+307) (+ x y) (/ (* y z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= 1e+307) {
		tmp = x + y;
	} else {
		tmp = (y * z) / t;
	}
	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 (((x + y) - (((z - t) * y) / (a - t))) <= 1d+307) then
        tmp = x + y
    else
        tmp = (y * z) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= 1e+307) {
		tmp = x + y;
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((x + y) - (((z - t) * y) / (a - t))) <= 1e+307:
		tmp = x + y
	else:
		tmp = (y * z) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t))) <= 1e+307)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(y * z) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x + y) - (((z - t) * y) / (a - t))) <= 1e+307)
		tmp = x + y;
	else
		tmp = (y * z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+307], N[(x + y), $MachinePrecision], N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 9.99999999999999986e306

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 86.0

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-+.f64 60.7

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

   10. Applied rewrites 60.7%

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

   if 9.99999999999999986e306 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 76.9%

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

   2. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-negate-rev N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 26.2

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

   4. Applied rewrites 26.2%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 18.5

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

   7. Applied rewrites 18.5%

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

Alternative 9: 60.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

2. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 93.6

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

4. Applied rewrites 93.6%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift--.f64 86.0

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in a around inf

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

   1. lower-+.f64 60.7

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

10. Applied rewrites 60.7%

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

Alternative 10: 19.5% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

2. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 93.6

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

4. Applied rewrites 93.6%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift--.f64 86.0

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in a around inf

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

   1. lower-+.f64 60.7

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

10. Applied rewrites 60.7%

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

11. Taylor expanded in x around 0

    \[\leadsto y \]
12. Step-by-step derivation

13. Applied rewrites 19.5%

    \[\leadsto y \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64
  (- (+ x y) (/ (* (- z t) y) (- a t))))