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

Percentage Accurate: 76.7% → 88.9%

Time: 3.8s

Alternatives: 11

Speedup: 0.9×

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: 76.7% 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: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{y \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{t - z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.2e+50)
   (+ x (/ (* y (- z a)) t))
   (if (<= t 2.4e+65)
     (fma (+ 1.0 (/ (- t z) (- a t))) y x)
     (fma (/ (- z a) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.2e+50) {
		tmp = x + ((y * (z - a)) / t);
	} else if (t <= 2.4e+65) {
		tmp = fma((1.0 + ((t - z) / (a - t))), y, x);
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.2e+50)
		tmp = Float64(x + Float64(Float64(y * Float64(z - a)) / t));
	elseif (t <= 2.4e+65)
		tmp = fma(Float64(1.0 + Float64(Float64(t - z) / Float64(a - t))), y, x);
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.2e+50], N[(x + N[(N[(y * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e+65], N[(N[(1.0 + N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.1999999999999999e50

   1. Initial program 57.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -4.1999999999999999e50 < t < 2.4000000000000002e65

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 2.4000000000000002e65 < t

   1. Initial program 57.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.1

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

   8. Applied rewrites 87.1%

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

Alternative 2: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ y t))) (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 INFINITY) (+ y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / t);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / t);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (y / t)
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(y / t))
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (y / t);
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or +inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 36.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 56.5

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

   11. Applied rewrites 56.5%

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

   12. Taylor expanded in x around 0

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

       1. lift-/.f64 41.5

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

   14. Applied rewrites 41.5%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < +inf.0

   1. Initial program 82.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 65.8

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

   5. Applied rewrites 65.8%

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

Alternative 3: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{y \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+65}:\\ \;\;\;\;\left(x + y\right) - y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.7e+50)
   (+ x (/ (* y (- z a)) t))
   (if (<= t 1.15e+65)
     (- (+ x y) (* y (/ z (- a t))))
     (fma (/ (- z a) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.7e+50) {
		tmp = x + ((y * (z - a)) / t);
	} else if (t <= 1.15e+65) {
		tmp = (x + y) - (y * (z / (a - t)));
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.7e+50)
		tmp = Float64(x + Float64(Float64(y * Float64(z - a)) / t));
	elseif (t <= 1.15e+65)
		tmp = Float64(Float64(x + y) - Float64(y * Float64(z / Float64(a - t))));
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.7e+50], N[(x + N[(N[(y * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+65], N[(N[(x + y), $MachinePrecision] - N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7e50

   1. Initial program 57.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -2.7e50 < t < 1.15e65

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if 1.15e65 < t

   1. Initial program 57.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.1

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

   8. Applied rewrites 87.1%

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

Alternative 4: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{+38}:\\ \;\;\;\;x + \frac{y \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-37}:\\ \;\;\;\;\left(x + y\right) - y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.56e+38)
   (+ x (/ (* y (- z a)) t))
   (if (<= t 4.6e-37) (- (+ x y) (* y (/ z a))) (fma (/ (- z a) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.56e+38) {
		tmp = x + ((y * (z - a)) / t);
	} else if (t <= 4.6e-37) {
		tmp = (x + y) - (y * (z / a));
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.56e+38)
		tmp = Float64(x + Float64(Float64(y * Float64(z - a)) / t));
	elseif (t <= 4.6e-37)
		tmp = Float64(Float64(x + y) - Float64(y * Float64(z / a)));
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.56e+38], N[(x + N[(N[(y * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-37], N[(N[(x + y), $MachinePrecision] - N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5599999999999999e38

   1. Initial program 58.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 77.0

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

   8. Applied rewrites 77.0%

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

   if -1.5599999999999999e38 < t < 4.5999999999999999e-37

   1. Initial program 92.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.9%

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

   if 4.5999999999999999e-37 < t

   1. Initial program 64.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 81.8

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

   8. Applied rewrites 81.8%

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

Alternative 5: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{+38}:\\ \;\;\;\;x + \frac{y \cdot \left(z - a\right)}{t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.56e+38)
   (+ x (/ (* y (- z a)) t))
   (if (<= t 4.6e-37) (fma (- 1.0 (/ z a)) y x) (fma (/ (- z a) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.56e+38) {
		tmp = x + ((y * (z - a)) / t);
	} else if (t <= 4.6e-37) {
		tmp = fma((1.0 - (z / a)), y, x);
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.56e+38)
		tmp = Float64(x + Float64(Float64(y * Float64(z - a)) / t));
	elseif (t <= 4.6e-37)
		tmp = fma(Float64(1.0 - Float64(z / a)), y, x);
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.56e+38], N[(x + N[(N[(y * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-37], N[(N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5599999999999999e38

   1. Initial program 58.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 77.0

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

   8. Applied rewrites 77.0%

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

   if -1.5599999999999999e38 < t < 4.5999999999999999e-37

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 81.9

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

   8. Applied rewrites 81.9%

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

   if 4.5999999999999999e-37 < t

   1. Initial program 64.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 81.8

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

   8. Applied rewrites 81.8%

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

Alternative 6: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{if}\;t \leq -1.56 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) y x)))
   (if (<= t -1.56e+38)
     t_1
     (if (<= t 4.6e-37) (fma (- 1.0 (/ z a)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - a) / t), y, x);
	double tmp;
	if (t <= -1.56e+38) {
		tmp = t_1;
	} else if (t <= 4.6e-37) {
		tmp = fma((1.0 - (z / a)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - a) / t), y, x)
	tmp = 0.0
	if (t <= -1.56e+38)
		tmp = t_1;
	elseif (t <= 4.6e-37)
		tmp = fma(Float64(1.0 - Float64(z / a)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -1.56e+38], t$95$1, If[LessEqual[t, 4.6e-37], N[(N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5599999999999999e38 or 4.5999999999999999e-37 < t

   1. Initial program 61.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.7

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

   8. Applied rewrites 83.7%

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

   if -1.5599999999999999e38 < t < 4.5999999999999999e-37

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 81.9

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

   8. Applied rewrites 81.9%

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

Alternative 7: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (/ z a)) y x)))
   (if (<= a -6.5e+62) t_1 (if (<= a 1.2e-78) (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 <= -6.5e+62) {
		tmp = t_1;
	} else if (a <= 1.2e-78) {
		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 <= -6.5e+62)
		tmp = t_1;
	elseif (a <= 1.2e-78)
		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, -6.5e+62], t$95$1, If[LessEqual[a, 1.2e-78], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000003e62 or 1.2e-78 < a

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 84.8

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

   8. Applied rewrites 84.8%

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

   if -6.5000000000000003e62 < a < 1.2e-78

   1. Initial program 74.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 76.9

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

   8. Applied rewrites 76.9%

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

Alternative 8: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.6e+62) (+ y x) (if (<= a 1.7e+97) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.6e+62) {
		tmp = y + x;
	} else if (a <= 1.7e+97) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.6e+62)
		tmp = Float64(y + x);
	elseif (a <= 1.7e+97)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.6e+62], N[(y + x), $MachinePrecision], If[LessEqual[a, 1.7e+97], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.59999999999999967e62 or 1.70000000000000005e97 < a

   1. Initial program 78.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -7.59999999999999967e62 < a < 1.70000000000000005e97

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\left(1 + \frac{t}{a - t}\right) - \frac{z}{a - t}\right)} \]
   4. 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. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift--.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 72.8

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

   8. Applied rewrites 72.8%

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

Alternative 9: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+33}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9.8e+45) x (if (<= t 1.15e+33) (+ y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.8e+45) {
		tmp = x;
	} else if (t <= 1.15e+33) {
		tmp = y + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-9.8d+45)) then
        tmp = x
    else if (t <= 1.15d+33) then
        tmp = y + x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.8e+45) {
		tmp = x;
	} else if (t <= 1.15e+33) {
		tmp = y + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -9.8e+45:
		tmp = x
	elif t <= 1.15e+33:
		tmp = y + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9.8e+45)
		tmp = x;
	elseif (t <= 1.15e+33)
		tmp = Float64(y + x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -9.8e+45)
		tmp = x;
	elseif (t <= 1.15e+33)
		tmp = y + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9.8e+45], x, If[LessEqual[t, 1.15e+33], N[(y + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -9.8000000000000004e45 or 1.15000000000000005e33 < t

   1. Initial program 58.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 62.6%

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

   if -9.8000000000000004e45 < t < 1.15000000000000005e33

   1. Initial program 91.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 63.9

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

   5. Applied rewrites 63.9%

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

Alternative 10: 53.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+55}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+183}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -9.5e+55) y (if (<= y 3.4e+183) x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -9.5e+55) {
		tmp = y;
	} else if (y <= 3.4e+183) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-9.5d+55)) then
        tmp = y
    else if (y <= 3.4d+183) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -9.5e+55) {
		tmp = y;
	} else if (y <= 3.4e+183) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -9.5e+55:
		tmp = y
	elif y <= 3.4e+183:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -9.5e+55)
		tmp = y;
	elseif (y <= 3.4e+183)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -9.5e+55)
		tmp = y;
	elseif (y <= 3.4e+183)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -9.5e+55], y, If[LessEqual[y, 3.4e+183], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -9.49999999999999989e55 or 3.4e183 < y

   1. Initial program 54.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 28.7%

      \[\leadsto y \]

   if -9.49999999999999989e55 < y < 3.4e183

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 63.9%

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

Alternative 11: 51.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 51.8%

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

Developer Target 1: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (< t_2 -1.3664970889390727e-7)
     t_1
     (if (< t_2 1.4754293444577233e-239)
       (/ (- (* y (- a z)) (* x t)) (- a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y + x) - (((z - t) * (1.0d0 / (a - t))) * y)
    t_2 = (x + y) - (((z - t) * y) / (a - t))
    if (t_2 < (-1.3664970889390727d-7)) then
        tmp = t_1
    else if (t_2 < 1.4754293444577233d-239) then
        tmp = ((y * (a - z)) - (x * t)) / (a - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y)
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 < -1.3664970889390727e-7:
		tmp = t_1
	elif t_2 < 1.4754293444577233e-239:
		tmp = ((y * (a - z)) - (x * t)) / (a - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * Float64(1.0 / Float64(a - t))) * y))
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = Float64(Float64(Float64(y * Float64(a - z)) - Float64(x * t)) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -1.3664970889390727e-7], t$95$1, If[Less[t$95$2, 1.4754293444577233e-239], N[(N[(N[(y * N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2025089 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -13664970889390727/100000000000000000000000) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 14754293444577233/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)))))

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