Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.8% → 99.9%

Time: 3.0s

Alternatives: 12

Speedup: 1.1×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma y 5.0 (* (+ (+ (fma 2.0 z y) y) t) x)))

C

double code(double x, double y, double z, double t) {
	return fma(y, 5.0, (((fma(2.0, z, y) + y) + t) * x));
}

Julia

function code(x, y, z, t)
	return fma(y, 5.0, Float64(Float64(Float64(fma(2.0, z, y) + y) + t) * x))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(N[(N[(N[(2.0 * z + y), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

   4. lift-+.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

   5. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

   6. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

   7. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right)} \]
4. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, \mathsf{fma}\left(2, z, t\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (fma 2.0 x 5.0) y (* (fma 2.0 z t) x)))

C

double code(double x, double y, double z, double t) {
	return fma(fma(2.0, x, 5.0), y, (fma(2.0, z, t) * x));
}

Julia

function code(x, y, z, t)
	return fma(fma(2.0, x, 5.0), y, Float64(fma(2.0, z, t) * x))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(2.0 * x + 5.0), $MachinePrecision] * y + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right) + y \cdot \left(5 + 2 \cdot x\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto y \cdot \left(5 + 2 \cdot x\right) + \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(5 + 2 \cdot x\right) \cdot y + \color{blue}{x} \cdot \left(t + 2 \cdot z\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(5 + 2 \cdot x, \color{blue}{y}, x \cdot \left(t + 2 \cdot z\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot x + 5, y, x \cdot \left(t + 2 \cdot z\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, x \cdot \left(t + 2 \cdot z\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, \left(t + 2 \cdot z\right) \cdot x\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, \left(t + 2 \cdot z\right) \cdot x\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, \left(2 \cdot z + t\right) \cdot x\right) \]

   9. lower-fma.f64 98.3

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, \mathsf{fma}\left(2, z, t\right) \cdot x\right) \]

4. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, 5\right), y, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{elif}\;t \leq 1050000:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(z + y\right), x, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.9e+54)
   (fma y 5.0 (* t x))
   (if (<= t 1050000.0)
     (fma (* 2.0 (+ z y)) x (* 5.0 y))
     (* (fma 2.0 (+ z y) t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.9e+54) {
		tmp = fma(y, 5.0, (t * x));
	} else if (t <= 1050000.0) {
		tmp = fma((2.0 * (z + y)), x, (5.0 * y));
	} else {
		tmp = fma(2.0, (z + y), t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.9e+54)
		tmp = fma(y, 5.0, Float64(t * x));
	elseif (t <= 1050000.0)
		tmp = fma(Float64(2.0 * Float64(z + y)), x, Float64(5.0 * y));
	else
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.9e+54], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1050000.0], N[(N[(2.0 * N[(z + y), $MachinePrecision]), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9000000000000001e54

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      7. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right)} \]

   4. Taylor expanded in t around inf

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

      1. associate-+l+ 79.8

         \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x\right) \]

      2. count-2-rev 79.8

         \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x\right) \]

      3. distribute-lft-in 79.8

         \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x\right) \]

   6. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t} \cdot x\right) \]

   if -1.9000000000000001e54 < t < 1.05e6

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(2 \cdot y + 2 \cdot z\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(2 \cdot y + 2 \cdot z\right) + \color{blue}{5 \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot y + 2 \cdot z\right) \cdot x + \color{blue}{5} \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot y + 2 \cdot z, \color{blue}{x}, 5 \cdot y\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(y + z\right), x, 5 \cdot y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(y + z\right), x, 5 \cdot y\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(z + y\right), x, 5 \cdot y\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(z + y\right), x, 5 \cdot y\right) \]

      8. lower-*.f64 92.2

         \[\leadsto \mathsf{fma}\left(2 \cdot \left(z + y\right), x, 5 \cdot y\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \left(z + y\right), x, 5 \cdot y\right)} \]

   if 1.05e6 < t

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, y + z, t\right) \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

      7. lower-+.f64 78.8

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

   4. Applied rewrites 78.8%

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

Alternative 4: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{if}\;x \leq -8 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ z y) t) x)))
   (if (<= x -8e-32) t_1 (if (<= x 6e-60) (fma (+ z z) x (* 5.0 y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (z + y), t) * x;
	double tmp;
	if (x <= -8e-32) {
		tmp = t_1;
	} else if (x <= 6e-60) {
		tmp = fma((z + z), x, (5.0 * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(z + y), t) * x)
	tmp = 0.0
	if (x <= -8e-32)
		tmp = t_1;
	elseif (x <= 6e-60)
		tmp = fma(Float64(z + z), x, Float64(5.0 * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -8e-32], t$95$1, If[LessEqual[x, 6e-60], N[(N[(z + z), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000045e-32 or 6.00000000000000038e-60 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, y + z, t\right) \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

      7. lower-+.f64 94.7

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -8.00000000000000045e-32 < x < 6.00000000000000038e-60

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      7. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right)} \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
   5. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right) \]

      2. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right) \]

      4. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(z + \color{blue}{z}\right) \cdot x\right) \]

      5. lower-+.f64 79.7

         \[\leadsto \mathsf{fma}\left(y, 5, \left(z + \color{blue}{z}\right) \cdot x\right) \]

   6. Applied rewrites 79.7%

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

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5 + \left(z + z\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + z\right) \cdot x + y \cdot 5} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(z + z\right) \cdot x} + y \cdot 5 \]

      4. *-commutative N/A

         \[\leadsto \left(z + z\right) \cdot x + \color{blue}{5 \cdot y} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z + z, x, 5 \cdot y\right)} \]

      6. lift-*.f64 79.7

         \[\leadsto \mathsf{fma}\left(z + z, x, \color{blue}{5 \cdot y}\right) \]

   8. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + z, x, 5 \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{if}\;x \leq -8 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ z y) t) x)))
   (if (<= x -8e-32) t_1 (if (<= x 6e-60) (fma y 5.0 (* (+ z z) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (z + y), t) * x;
	double tmp;
	if (x <= -8e-32) {
		tmp = t_1;
	} else if (x <= 6e-60) {
		tmp = fma(y, 5.0, ((z + z) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(z + y), t) * x)
	tmp = 0.0
	if (x <= -8e-32)
		tmp = t_1;
	elseif (x <= 6e-60)
		tmp = fma(y, 5.0, Float64(Float64(z + z) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -8e-32], t$95$1, If[LessEqual[x, 6e-60], N[(y * 5.0 + N[(N[(z + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000045e-32 or 6.00000000000000038e-60 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, y + z, t\right) \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

      7. lower-+.f64 94.7

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -8.00000000000000045e-32 < x < 6.00000000000000038e-60

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      7. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right)} \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
   5. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right) \]

      2. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right) \]

      4. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(z + \color{blue}{z}\right) \cdot x\right) \]

      5. lower-+.f64 79.7

         \[\leadsto \mathsf{fma}\left(y, 5, \left(z + \color{blue}{z}\right) \cdot x\right) \]

   6. Applied rewrites 79.7%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z + z\right)} \cdot x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{if}\;x \leq -1.26 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ z y) t) x)))
   (if (<= x -1.26e-32) t_1 (if (<= x 3.5e-107) (fma y 5.0 (* t x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (z + y), t) * x;
	double tmp;
	if (x <= -1.26e-32) {
		tmp = t_1;
	} else if (x <= 3.5e-107) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(z + y), t) * x)
	tmp = 0.0
	if (x <= -1.26e-32)
		tmp = t_1;
	elseif (x <= 3.5e-107)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.26e-32], t$95$1, If[LessEqual[x, 3.5e-107], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2599999999999999e-32 or 3.49999999999999985e-107 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, y + z, t\right) \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

      7. lower-+.f64 92.3

         \[\leadsto \mathsf{fma}\left(2, z + y, t\right) \cdot x \]

   4. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -1.2599999999999999e-32 < x < 3.49999999999999985e-107

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      7. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right)} \]

   4. Taylor expanded in t around inf

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

      1. associate-+l+ 81.6

         \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x\right) \]

      2. count-2-rev 81.6

         \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x\right) \]

      3. distribute-lft-in 81.6

         \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x\right) \]

   6. Applied rewrites 81.6%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t} \cdot x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 245000000000:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -1.6e+94)
     t_1
     (if (<= y 245000000000.0) (* (fma 2.0 z t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -1.6e+94) {
		tmp = t_1;
	} else if (y <= 245000000000.0) {
		tmp = fma(2.0, z, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -1.6e+94)
		tmp = t_1;
	elseif (y <= 245000000000.0)
		tmp = Float64(fma(2.0, z, t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.6e+94], t$95$1, If[LessEqual[y, 245000000000.0], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000007e94 or 2.45e11 < y

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(5 + 2 \cdot x\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(5 + 2 \cdot x\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot x + 5\right) \cdot y \]

      4. lower-fma.f64 80.4

         \[\leadsto \mathsf{fma}\left(2, x, 5\right) \cdot y \]

   4. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -1.60000000000000007e94 < y < 2.45e11

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + 2 \cdot z\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + 2 \cdot z\right) \cdot \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot z + t\right) \cdot x \]

      4. lower-fma.f64 76.7

         \[\leadsto \mathsf{fma}\left(2, z, t\right) \cdot x \]

   4. Applied rewrites 76.7%

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

Alternative 8: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + x\right) \cdot z\\ \mathbf{if}\;z \leq -8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ x x) z)))
   (if (<= z -8e+80) t_1 (if (<= z 4.4e+174) (* (fma 2.0 x 5.0) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + x) * z;
	double tmp;
	if (z <= -8e+80) {
		tmp = t_1;
	} else if (z <= 4.4e+174) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + x) * z)
	tmp = 0.0
	if (z <= -8e+80)
		tmp = t_1;
	elseif (z <= 4.4e+174)
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -8e+80], t$95$1, If[LessEqual[z, 4.4e+174], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e80 or 4.40000000000000039e174 < z

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{z} \]

      3. count-2-rev N/A

         \[\leadsto \left(x + x\right) \cdot z \]

      4. lower-+.f64 63.7

         \[\leadsto \left(x + x\right) \cdot z \]

   4. Applied rewrites 63.7%

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

   if -8e80 < z < 4.40000000000000039e174

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(5 + 2 \cdot x\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(5 + 2 \cdot x\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot x + 5\right) \cdot y \]

      4. lower-fma.f64 54.8

         \[\leadsto \mathsf{fma}\left(2, x, 5\right) \cdot y \]

   4. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+66}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+48}:\\ \;\;\;\;\left(x + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.4e+66) (* t x) (if (<= t 1.1e+48) (* (+ x x) z) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.4e+66) {
		tmp = t * x;
	} else if (t <= 1.1e+48) {
		tmp = (x + x) * z;
	} else {
		tmp = t * 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)
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) :: tmp
    if (t <= (-6.4d+66)) then
        tmp = t * x
    else if (t <= 1.1d+48) then
        tmp = (x + x) * z
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.4e+66) {
		tmp = t * x;
	} else if (t <= 1.1e+48) {
		tmp = (x + x) * z;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.4e+66:
		tmp = t * x
	elif t <= 1.1e+48:
		tmp = (x + x) * z
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.4e+66)
		tmp = Float64(t * x);
	elseif (t <= 1.1e+48)
		tmp = Float64(Float64(x + x) * z);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.4e+66)
		tmp = t * x;
	elseif (t <= 1.1e+48)
		tmp = (x + x) * z;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.4e+66], N[(t * x), $MachinePrecision], If[LessEqual[t, 1.1e+48], N[(N[(x + x), $MachinePrecision] * z), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.3999999999999999e66 or 1.1e48 < t

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 60.9

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

   4. Applied rewrites 60.9%

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

   if -6.3999999999999999e66 < t < 1.1e48

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{z} \]

      3. count-2-rev N/A

         \[\leadsto \left(x + x\right) \cdot z \]

      4. lower-+.f64 37.1

         \[\leadsto \left(x + x\right) \cdot z \]

   4. Applied rewrites 37.1%

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

Alternative 10: 46.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + x\right) \cdot y\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+186}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -2.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+79}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+253}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ x x) y)))
   (if (<= x -1.6e+186)
     (* t x)
     (if (<= x -2.5)
       t_1
       (if (<= x 3.8e-97)
         (* 5.0 y)
         (if (<= x 8.8e+79) (* t x) (if (<= x 4e+253) t_1 (* t x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + x) * y;
	double tmp;
	if (x <= -1.6e+186) {
		tmp = t * x;
	} else if (x <= -2.5) {
		tmp = t_1;
	} else if (x <= 3.8e-97) {
		tmp = 5.0 * y;
	} else if (x <= 8.8e+79) {
		tmp = t * x;
	} else if (x <= 4e+253) {
		tmp = t_1;
	} else {
		tmp = t * 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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x + x) * y
    if (x <= (-1.6d+186)) then
        tmp = t * x
    else if (x <= (-2.5d0)) then
        tmp = t_1
    else if (x <= 3.8d-97) then
        tmp = 5.0d0 * y
    else if (x <= 8.8d+79) then
        tmp = t * x
    else if (x <= 4d+253) then
        tmp = t_1
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + x) * y;
	double tmp;
	if (x <= -1.6e+186) {
		tmp = t * x;
	} else if (x <= -2.5) {
		tmp = t_1;
	} else if (x <= 3.8e-97) {
		tmp = 5.0 * y;
	} else if (x <= 8.8e+79) {
		tmp = t * x;
	} else if (x <= 4e+253) {
		tmp = t_1;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + x) * y
	tmp = 0
	if x <= -1.6e+186:
		tmp = t * x
	elif x <= -2.5:
		tmp = t_1
	elif x <= 3.8e-97:
		tmp = 5.0 * y
	elif x <= 8.8e+79:
		tmp = t * x
	elif x <= 4e+253:
		tmp = t_1
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + x) * y)
	tmp = 0.0
	if (x <= -1.6e+186)
		tmp = Float64(t * x);
	elseif (x <= -2.5)
		tmp = t_1;
	elseif (x <= 3.8e-97)
		tmp = Float64(5.0 * y);
	elseif (x <= 8.8e+79)
		tmp = Float64(t * x);
	elseif (x <= 4e+253)
		tmp = t_1;
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + x) * y;
	tmp = 0.0;
	if (x <= -1.6e+186)
		tmp = t * x;
	elseif (x <= -2.5)
		tmp = t_1;
	elseif (x <= 3.8e-97)
		tmp = 5.0 * y;
	elseif (x <= 8.8e+79)
		tmp = t * x;
	elseif (x <= 4e+253)
		tmp = t_1;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, -1.6e+186], N[(t * x), $MachinePrecision], If[LessEqual[x, -2.5], t$95$1, If[LessEqual[x, 3.8e-97], N[(5.0 * y), $MachinePrecision], If[LessEqual[x, 8.8e+79], N[(t * x), $MachinePrecision], If[LessEqual[x, 4e+253], t$95$1, N[(t * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e186 or 3.8000000000000001e-97 < x < 8.7999999999999996e79 or 3.9999999999999997e253 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.6

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

   4. Applied rewrites 37.6%

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

   if -1.6e186 < x < -2.5 or 8.7999999999999996e79 < x < 3.9999999999999997e253

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      7. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \left(\left(\mathsf{fma}\left(2, z, y\right) + y\right) + t\right) \cdot x\right)} \]

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      2. associate-+l+ N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      4. count-2-rev N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      5. associate-+l+ N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      6. associate-+l+ N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{y} \cdot \left(5 + 2 \cdot x\right) \]

      8. *-commutative N/A

         \[\leadsto \left(5 + 2 \cdot x\right) \cdot \color{blue}{y} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 + 2 \cdot x\right) \cdot \color{blue}{y} \]

      10. +-commutative N/A

          \[\leadsto \left(2 \cdot x + 5\right) \cdot y \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot 2 + 5\right) \cdot y \]

      12. lower-fma.f64 36.6

          \[\leadsto \mathsf{fma}\left(x, 2, 5\right) \cdot y \]

   6. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2, 5\right) \cdot y} \]

   7. Taylor expanded in x around inf

      \[\leadsto \left(2 \cdot x\right) \cdot y \]
   8. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \left(x + x\right) \cdot y \]

      2. lower-+.f64 35.9

         \[\leadsto \left(x + x\right) \cdot y \]

   9. Applied rewrites 35.9%

      \[\leadsto \left(x + x\right) \cdot y \]

   if -2.5 < x < 3.8000000000000001e-97

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 60.4

         \[\leadsto 5 \cdot \color{blue}{y} \]

   4. Applied rewrites 60.4%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-32}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -8e-32) (* t x) (if (<= x 3.8e-97) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8e-32) {
		tmp = t * x;
	} else if (x <= 3.8e-97) {
		tmp = 5.0 * y;
	} else {
		tmp = t * 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)
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) :: tmp
    if (x <= (-8d-32)) then
        tmp = t * x
    else if (x <= 3.8d-97) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8e-32) {
		tmp = t * x;
	} else if (x <= 3.8e-97) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -8e-32:
		tmp = t * x
	elif x <= 3.8e-97:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -8e-32)
		tmp = Float64(t * x);
	elseif (x <= 3.8e-97)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -8e-32)
		tmp = t * x;
	elseif (x <= 3.8e-97)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -8e-32], N[(t * x), $MachinePrecision], If[LessEqual[x, 3.8e-97], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000045e-32 or 3.8000000000000001e-97 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.8

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

   4. Applied rewrites 37.8%

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

   if -8.00000000000000045e-32 < x < 3.8000000000000001e-97

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 62.4

         \[\leadsto 5 \cdot \color{blue}{y} \]

   4. Applied rewrites 62.4%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ 5 \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 5.0 y))

C

double code(double x, double y, double z, double t) {
	return 5.0 * y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 5.0d0 * y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 5.0 * y;
}

Python

def code(x, y, z, t):
	return 5.0 * y

Julia

function code(x, y, z, t)
	return Float64(5.0 * y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 5.0 * y;
end

Wolfram

code[x_, y_, z_, t_] := N[(5.0 * y), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
3. Step-by-step derivation

   1. lower-*.f64 29.9

      \[\leadsto 5 \cdot \color{blue}{y} \]

4. Applied rewrites 29.9%

   \[\leadsto \color{blue}{5 \cdot y} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))