Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 12

Speedup: 1.2×

• 35.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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)
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) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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)
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) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (fma (- y z) (- t x) x))

C

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

Julia

function code(x, y, z, t)
	return fma(Float64(y - z), Float64(t - x), x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -1.14 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(-t, z, x\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -1.14e-60)
     t_1
     (if (<= y 5e-100)
       (fma (- t) z x)
       (if (<= y 3.1e+27) (* (- x t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -1.14e-60) {
		tmp = t_1;
	} else if (y <= 5e-100) {
		tmp = fma(-t, z, x);
	} else if (y <= 3.1e+27) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -1.14e-60)
		tmp = t_1;
	elseif (y <= 5e-100)
		tmp = fma(Float64(-t), z, x);
	elseif (y <= 3.1e+27)
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.14e-60], t$95$1, If[LessEqual[y, 5e-100], N[((-t) * z + x), $MachinePrecision], If[LessEqual[y, 3.1e+27], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.14000000000000001e-60 or 3.09999999999999996e27 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -1.14000000000000001e-60 < y < 5.0000000000000001e-100

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 96.7

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.6%

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

   if 5.0000000000000001e-100 < y < 3.09999999999999996e27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.5%

      \[\leadsto \left(x - t\right) \cdot \color{blue}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.14 \cdot 10^{-60}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(-t, z, x\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -1.14 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-100}:\\ \;\;\;\;x - t \cdot z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -1.14e-60)
     t_1
     (if (<= y 5e-100) (- x (* t z)) (if (<= y 3.1e+27) (* (- x t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -1.14e-60) {
		tmp = t_1;
	} else if (y <= 5e-100) {
		tmp = x - (t * z);
	} else if (y <= 3.1e+27) {
		tmp = (x - t) * z;
	} 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)
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 = (t - x) * y
    if (y <= (-1.14d-60)) then
        tmp = t_1
    else if (y <= 5d-100) then
        tmp = x - (t * z)
    else if (y <= 3.1d+27) then
        tmp = (x - t) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -1.14e-60) {
		tmp = t_1;
	} else if (y <= 5e-100) {
		tmp = x - (t * z);
	} else if (y <= 3.1e+27) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t - x) * y
	tmp = 0
	if y <= -1.14e-60:
		tmp = t_1
	elif y <= 5e-100:
		tmp = x - (t * z)
	elif y <= 3.1e+27:
		tmp = (x - t) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -1.14e-60)
		tmp = t_1;
	elseif (y <= 5e-100)
		tmp = Float64(x - Float64(t * z));
	elseif (y <= 3.1e+27)
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t - x) * y;
	tmp = 0.0;
	if (y <= -1.14e-60)
		tmp = t_1;
	elseif (y <= 5e-100)
		tmp = x - (t * z);
	elseif (y <= 3.1e+27)
		tmp = (x - t) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.14e-60], t$95$1, If[LessEqual[y, 5e-100], N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+27], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.14000000000000001e-60 or 3.09999999999999996e27 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -1.14000000000000001e-60 < y < 5.0000000000000001e-100

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.6%

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

   if 5.0000000000000001e-100 < y < 3.09999999999999996e27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.5%

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

Alternative 4: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-134}:\\ \;\;\;\;x \cdot z + x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -5.6e-52)
     t_1
     (if (<= y 5.7e-134)
       (+ (* x z) x)
       (if (<= y 3.1e+27) (* (- x t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -5.6e-52) {
		tmp = t_1;
	} else if (y <= 5.7e-134) {
		tmp = (x * z) + x;
	} else if (y <= 3.1e+27) {
		tmp = (x - t) * z;
	} 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)
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 = (t - x) * y
    if (y <= (-5.6d-52)) then
        tmp = t_1
    else if (y <= 5.7d-134) then
        tmp = (x * z) + x
    else if (y <= 3.1d+27) then
        tmp = (x - t) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -5.6e-52) {
		tmp = t_1;
	} else if (y <= 5.7e-134) {
		tmp = (x * z) + x;
	} else if (y <= 3.1e+27) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t - x) * y
	tmp = 0
	if y <= -5.6e-52:
		tmp = t_1
	elif y <= 5.7e-134:
		tmp = (x * z) + x
	elif y <= 3.1e+27:
		tmp = (x - t) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -5.6e-52)
		tmp = t_1;
	elseif (y <= 5.7e-134)
		tmp = Float64(Float64(x * z) + x);
	elseif (y <= 3.1e+27)
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t - x) * y;
	tmp = 0.0;
	if (y <= -5.6e-52)
		tmp = t_1;
	elseif (y <= 5.7e-134)
		tmp = (x * z) + x;
	elseif (y <= 3.1e+27)
		tmp = (x - t) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -5.6e-52], t$95$1, If[LessEqual[y, 5.7e-134], N[(N[(x * z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.1e+27], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.59999999999999989e-52 or 3.09999999999999996e27 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -5.59999999999999989e-52 < y < 5.70000000000000029e-134

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.4%

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

   10. Applied rewrites 64.4%

       \[\leadsto x \cdot z + x \]

   if 5.70000000000000029e-134 < y < 3.09999999999999996e27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.8%

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

Alternative 5: 54.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-295}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 1950:\\ \;\;\;\;x \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -1.9e+74)
     t_1
     (if (<= z -7e-295) (* t y) (if (<= z 1950.0) (+ (* x z) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -1.9e+74) {
		tmp = t_1;
	} else if (z <= -7e-295) {
		tmp = t * y;
	} else if (z <= 1950.0) {
		tmp = (x * z) + x;
	} 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)
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 - t) * z
    if (z <= (-1.9d+74)) then
        tmp = t_1
    else if (z <= (-7d-295)) then
        tmp = t * y
    else if (z <= 1950.0d0) then
        tmp = (x * z) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -1.9e+74) {
		tmp = t_1;
	} else if (z <= -7e-295) {
		tmp = t * y;
	} else if (z <= 1950.0) {
		tmp = (x * z) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - t) * z
	tmp = 0
	if z <= -1.9e+74:
		tmp = t_1
	elif z <= -7e-295:
		tmp = t * y
	elif z <= 1950.0:
		tmp = (x * z) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -1.9e+74)
		tmp = t_1;
	elseif (z <= -7e-295)
		tmp = Float64(t * y);
	elseif (z <= 1950.0)
		tmp = Float64(Float64(x * z) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - t) * z;
	tmp = 0.0;
	if (z <= -1.9e+74)
		tmp = t_1;
	elseif (z <= -7e-295)
		tmp = t * y;
	elseif (z <= 1950.0)
		tmp = (x * z) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.9e+74], t$95$1, If[LessEqual[z, -7e-295], N[(t * y), $MachinePrecision], If[LessEqual[z, 1950.0], N[(N[(x * z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e74 or 1950 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.5%

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

   if -1.8999999999999999e74 < z < -6.99999999999999977e-295

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.5%

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

   if -6.99999999999999977e-295 < z < 1950

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.6%

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

   10. Applied rewrites 45.6%

       \[\leadsto x \cdot z + x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-295}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 1950:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -1.9e+74)
     t_1
     (if (<= z -7e-295) (* t y) (if (<= z 1950.0) (fma x z x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -1.9e+74) {
		tmp = t_1;
	} else if (z <= -7e-295) {
		tmp = t * y;
	} else if (z <= 1950.0) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -1.9e+74)
		tmp = t_1;
	elseif (z <= -7e-295)
		tmp = Float64(t * y);
	elseif (z <= 1950.0)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.9e+74], t$95$1, If[LessEqual[z, -7e-295], N[(t * y), $MachinePrecision], If[LessEqual[z, 1950.0], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e74 or 1950 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.5%

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

   if -1.8999999999999999e74 < z < -6.99999999999999977e-295

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.5%

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

   if -6.99999999999999977e-295 < z < 1950

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.6%

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

Alternative 7: 50.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+54}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.06e+99)
   (* t y)
   (if (<= y -1.65e+54) (* (- x) y) (if (<= y 2.3e+25) (fma x z x) (* t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = t * y;
	} else if (y <= -1.65e+54) {
		tmp = -x * y;
	} else if (y <= 2.3e+25) {
		tmp = fma(x, z, x);
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.06e+99)
		tmp = Float64(t * y);
	elseif (y <= -1.65e+54)
		tmp = Float64(Float64(-x) * y);
	elseif (y <= 2.3e+25)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.06e+99], N[(t * y), $MachinePrecision], If[LessEqual[y, -1.65e+54], N[((-x) * y), $MachinePrecision], If[LessEqual[y, 2.3e+25], N[(x * z + x), $MachinePrecision], N[(t * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05999999999999999e99 or 2.2999999999999998e25 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.9%

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

   if -1.05999999999999999e99 < y < -1.65e54

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -1.65e54 < y < 2.2999999999999998e25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.4%

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

Alternative 8: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+74} \lor \neg \left(z \leq 9.6 \cdot 10^{+106}\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.9e+74) (not (<= z 9.6e+106)))
   (* (- x t) z)
   (fma (- t x) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.9e+74) || !(z <= 9.6e+106)) {
		tmp = (x - t) * z;
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.9e+74) || !(z <= 9.6e+106))
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.9e+74], N[Not[LessEqual[z, 9.6e+106]], $MachinePrecision]], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8999999999999999e74 or 9.6000000000000002e106 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 84.2%

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

   if -1.8999999999999999e74 < z < 9.6000000000000002e106

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+74} \lor \neg \left(z \leq 9.6 \cdot 10^{+106}\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.6e-52)
   (fma (- t x) y x)
   (if (<= y 3.1e+27) (fma (- x t) z x) (* (- t x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.6e-52) {
		tmp = fma((t - x), y, x);
	} else if (y <= 3.1e+27) {
		tmp = fma((x - t), z, x);
	} else {
		tmp = (t - x) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.6e-52)
		tmp = fma(Float64(t - x), y, x);
	elseif (y <= 3.1e+27)
		tmp = fma(Float64(x - t), z, x);
	else
		tmp = Float64(Float64(t - x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.6e-52], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 3.1e+27], N[(N[(x - t), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.59999999999999989e-52

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -5.59999999999999989e-52 < y < 3.09999999999999996e27

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if 3.09999999999999996e27 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 81.1

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

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\left(t - x\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{+59} \lor \neg \left(y \leq 2.3 \cdot 10^{+25}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -7.1e+59) (not (<= y 2.3e+25))) (* t y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7.1e+59) || !(y <= 2.3e+25)) {
		tmp = t * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -7.1e+59) || !(y <= 2.3e+25))
		tmp = Float64(t * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -7.1e+59], N[Not[LessEqual[y, 2.3e+25]], $MachinePrecision]], N[(t * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.10000000000000003e59 or 2.2999999999999998e25 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.8%

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

   if -7.10000000000000003e59 < y < 2.2999999999999998e25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.1%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{+59} \lor \neg \left(y \leq 2.3 \cdot 10^{+25}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+166} \lor \neg \left(z \leq 2.3 \cdot 10^{+149}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3e+166) (not (<= z 2.3e+149))) (* x z) (* t y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3e+166) || !(z <= 2.3e+149)) {
		tmp = x * z;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 ((z <= (-3d+166)) .or. (.not. (z <= 2.3d+149))) then
        tmp = x * z
    else
        tmp = t * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3e+166) || !(z <= 2.3e+149)) {
		tmp = x * z;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3e+166) or not (z <= 2.3e+149):
		tmp = x * z
	else:
		tmp = t * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3e+166) || !(z <= 2.3e+149))
		tmp = Float64(x * z);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3e+166) || ~((z <= 2.3e+149)))
		tmp = x * z;
	else
		tmp = t * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3e+166], N[Not[LessEqual[z, 2.3e+149]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(t * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999998e166 or 2.2999999999999998e149 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 49.3%

       \[\leadsto x \cdot z \]

   if -2.99999999999999998e166 < z < 2.2999999999999998e149

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.8%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+166} \lor \neg \left(z \leq 2.3 \cdot 10^{+149}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 47.6

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

5. Applied rewrites 47.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 30.0%

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

Developer Target 1: 96.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 + ((t * (y - z)) + (-x * (y - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (+ (* t (- y z)) (* (- x) (- y z)))))

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