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

Percentage Accurate: 77.5% → 94.2%

Time: 4.1s

Alternatives: 9

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: 77.5% 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: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\frac{t}{a - t} + 1\right) + \frac{x}{y}\right) - \frac{z}{a - t}\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{t - z}{a - t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_1 (- INFINITY))
     (* (- (+ (+ (/ t (- a t)) 1.0) (/ x y)) (/ z (- a t))) y)
     (if (<= t_1 -4e-178)
       t_1
       (if (<= t_1 0.0)
         (fma (/ (- z a) t) y x)
         (fma (+ 1.0 (/ (- t z) (- a t))) y x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((((t / (a - t)) + 1.0) + (x / y)) - (z / (a - t))) * y;
	} else if (t_1 <= -4e-178) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(((z - a) / t), y, x);
	} else {
		tmp = fma((1.0 + ((t - z) / (a - t))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(t / Float64(a - t)) + 1.0) + Float64(x / y)) - Float64(z / Float64(a - t))) * y);
	elseif (t_1 <= -4e-178)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	else
		tmp = fma(Float64(1.0 + Float64(Float64(t - z) / Float64(a - t))), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -4e-178], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(1.0 + N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 36.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -3.9999999999999998e-178

   1. Initial program 98.3%

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

   if -3.9999999999999998e-178 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 22.0%

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

   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. 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 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.7

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

   7. Applied rewrites 89.7%

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 83.1%

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

   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. 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 93.1

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

   4. Applied rewrites 93.1%

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

Alternative 2: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 + \frac{t - z}{a - t}, y, x\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-251}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;x + \mathsf{fma}\left(-1, \frac{a \cdot y}{t}, \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (+ 1.0 (/ (- t z) (- a t))) y x))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 -2e-251)
     t_1
     (if (<= t_2 0.0) (+ x (fma -1.0 (/ (* a y) t) (/ (* y z) t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((1.0 + ((t - z) / (a - t))), y, x);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -2e-251) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = x + fma(-1.0, ((a * y) / t), ((y * z) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(1.0 + Float64(Float64(t - z) / Float64(a - t))), y, x)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e-251)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(x + fma(-1.0, Float64(Float64(a * y) / t), Float64(Float64(y * z) / 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[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $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, -2e-251], t$95$1, If[LessEqual[t$95$2, 0.0], N[(x + N[(-1.0 * N[(N[(a * y), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $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))) < -2.00000000000000003e-251 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 83.0%

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

   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. 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 92.8

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

   4. Applied rewrites 92.8%

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

   if -2.00000000000000003e-251 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 8.6%

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

   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. 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 47.7

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

   4. Applied rewrites 47.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-/.f64 N/A

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

   6. Applied rewrites 47.7%

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

   7. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 96.9

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

   9. Applied rewrites 96.9%

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

   10. Taylor expanded in z around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 96.9

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

   12. Applied rewrites 96.9%

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

Alternative 3: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 + \frac{t - z}{a - t}, y, x\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-251}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;x + \frac{y \cdot \left(z - a\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (+ 1.0 (/ (- t z) (- a t))) y x))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 -2e-251) t_1 (if (<= t_2 0.0) (+ x (/ (* y (- z a)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((1.0 + ((t - z) / (a - t))), y, x);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -2e-251) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = x + ((y * (z - a)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(1.0 + Float64(Float64(t - z) / Float64(a - t))), y, x)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e-251)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(x + Float64(Float64(y * Float64(z - 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[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $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, -2e-251], t$95$1, If[LessEqual[t$95$2, 0.0], N[(x + N[(N[(y * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $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))) < -2.00000000000000003e-251 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 83.0%

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

   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. 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 92.8

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

   4. Applied rewrites 92.8%

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

   if -2.00000000000000003e-251 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 8.6%

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

   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. 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 47.7

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in t around -inf

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(z - a\right)}{t}} \]
   6. 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 96.9

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

   7. Applied rewrites 96.9%

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

Alternative 4: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) y x)))
   (if (<= t -1.4e-10) t_1 (if (<= t 1.8e-38) (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 <= -1.4e-10) {
		tmp = t_1;
	} else if (t <= 1.8e-38) {
		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 <= -1.4e-10)
		tmp = t_1;
	elseif (t <= 1.8e-38)
		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, -1.4e-10], t$95$1, If[LessEqual[t, 1.8e-38], 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 < -1.40000000000000008e-10 or 1.8e-38 < t

   1. Initial program 64.2%

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

   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. 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

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in t around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if -1.40000000000000008e-10 < t < 1.8e-38

   1. Initial program 93.1%

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

   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. 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

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

   4. Applied rewrites 94.2%

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

   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 84.0

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

   7. Applied rewrites 84.0%

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

Alternative 5: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{if}\;a \leq -0.25:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (/ z a)) y x)))
   (if (<= a -0.25) t_1 (if (<= a 1.15e+31) (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 <= -0.25) {
		tmp = t_1;
	} else if (a <= 1.15e+31) {
		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 <= -0.25)
		tmp = t_1;
	elseif (a <= 1.15e+31)
		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, -0.25], t$95$1, If[LessEqual[a, 1.15e+31], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.25 or 1.15e31 < a

   1. Initial program 79.4%

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

   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. 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 93.3

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

   4. Applied rewrites 93.3%

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

   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 86.4

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

   7. Applied rewrites 86.4%

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

   if -0.25 < a < 1.15e31

   1. Initial program 75.6%

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

   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. 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 85.8

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

   4. Applied rewrites 85.8%

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

   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 77.1

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

   7. Applied rewrites 77.1%

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

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+162}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.6e+162) (+ y x) (if (<= a 1.15e+31) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.6e+162) {
		tmp = y + x;
	} else if (a <= 1.15e+31) {
		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 <= -3.6e+162)
		tmp = Float64(y + x);
	elseif (a <= 1.15e+31)
		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, -3.6e+162], N[(y + x), $MachinePrecision], If[LessEqual[a, 1.15e+31], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.59999999999999994e162 or 1.15e31 < a

   1. Initial program 79.5%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   if -3.59999999999999994e162 < a < 1.15e31

   1. Initial program 76.4%

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

   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. 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 86.8

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

   4. Applied rewrites 86.8%

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

   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 71.9

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

   7. Applied rewrites 71.9%

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

Alternative 7: 63.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-27}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 45000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.5e-27) (+ y x) (if (<= a 45000000000000.0) x (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.5e-27) {
		tmp = y + x;
	} else if (a <= 45000000000000.0) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-7.5d-27)) then
        tmp = y + x
    else if (a <= 45000000000000.0d0) then
        tmp = x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.5e-27) {
		tmp = y + x;
	} else if (a <= 45000000000000.0) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -7.5e-27:
		tmp = y + x
	elif a <= 45000000000000.0:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.5e-27)
		tmp = Float64(y + x);
	elseif (a <= 45000000000000.0)
		tmp = x;
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -7.5e-27)
		tmp = y + x;
	elseif (a <= 45000000000000.0)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.5e-27], N[(y + x), $MachinePrecision], If[LessEqual[a, 45000000000000.0], x, N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.50000000000000029e-27 or 4.5e13 < a

   1. Initial program 79.6%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 75.8

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

   4. Applied rewrites 75.8%

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

   if -7.50000000000000029e-27 < a < 4.5e13

   1. Initial program 75.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.3%

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

Alternative 8: 53.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-270}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-126}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.5e-270) x (if (<= x 1.1e-126) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.5e-270) {
		tmp = x;
	} else if (x <= 1.1e-126) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.5d-270)) then
        tmp = x
    else if (x <= 1.1d-126) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.5e-270) {
		tmp = x;
	} else if (x <= 1.1e-126) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.5e-270:
		tmp = x
	elif x <= 1.1e-126:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.5e-270)
		tmp = x;
	elseif (x <= 1.1e-126)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.5e-270)
		tmp = x;
	elseif (x <= 1.1e-126)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.5e-270], x, If[LessEqual[x, 1.1e-126], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000006e-270 or 1.10000000000000007e-126 < x

   1. Initial program 79.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 58.3%

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

   if -1.50000000000000006e-270 < x < 1.10000000000000007e-126

   1. Initial program 69.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y - \frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. 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 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 30.9%

      \[\leadsto y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.4% 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 77.5%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 51.4%

   \[\leadsto \color{blue}{x} \]
5. 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 2025095 
(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))))