Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 10

Speedup: 1.2×

• 34.2% 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: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ t_2 := \mathsf{fma}\left(-y, x, x\right)\\ \mathbf{if}\;z \leq -780:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-190}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-294}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;z \leq 940000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)) (t_2 (fma (- y) x x)))
   (if (<= z -780.0)
     t_1
     (if (<= z -4.5e-190)
       t_2
       (if (<= z 9e-294) (* (- t x) y) (if (<= z 940000.0) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double t_2 = fma(-y, x, x);
	double tmp;
	if (z <= -780.0) {
		tmp = t_1;
	} else if (z <= -4.5e-190) {
		tmp = t_2;
	} else if (z <= 9e-294) {
		tmp = (t - x) * y;
	} else if (z <= 940000.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	t_2 = fma(Float64(-y), x, x)
	tmp = 0.0
	if (z <= -780.0)
		tmp = t_1;
	elseif (z <= -4.5e-190)
		tmp = t_2;
	elseif (z <= 9e-294)
		tmp = Float64(Float64(t - x) * y);
	elseif (z <= 940000.0)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[((-y) * x + x), $MachinePrecision]}, If[LessEqual[z, -780.0], t$95$1, If[LessEqual[z, -4.5e-190], t$95$2, If[LessEqual[z, 9e-294], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 940000.0], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -780 or 9.4e5 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

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

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

       3. mul-1-neg N/A

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

       4. mul-1-neg N/A

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

       5. *-rgt-identity N/A

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

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

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

       9. distribute-lft-neg-in N/A

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

       10. mul-1-neg N/A

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

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

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

       12. *-rgt-identity N/A

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

       13. distribute-neg-in N/A

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

       14. +-commutative N/A

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

       15. mul-1-neg N/A

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

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

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

       17. metadata-eval N/A

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

       18. *-lft-identity N/A

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

       19. mul-1-neg N/A

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

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

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

       21. *-commutative N/A

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

       22. *-commutative N/A

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

   11. Applied rewrites 83.1%

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

   if -780 < z < -4.50000000000000021e-190 or 8.99999999999999963e-294 < z < 9.4e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-in N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

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

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.8%

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

   if -4.50000000000000021e-190 < z < 8.99999999999999963e-294

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 3: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-11} \lor \neg \left(z \leq 1.45 \cdot 10^{-38}\right):\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.4e-11) (not (<= z 1.45e-38)))
   (fma (- x t) z x)
   (fma (- t x) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e-11) || !(z <= 1.45e-38)) {
		tmp = fma((x - t), z, x);
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.4e-11) || !(z <= 1.45e-38))
		tmp = fma(Float64(x - t), z, x);
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.4e-11], N[Not[LessEqual[z, 1.45e-38]], $MachinePrecision]], N[(N[(x - t), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999999e-11 or 1.44999999999999997e-38 < 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. +-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. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 82.5

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

   5. Applied rewrites 82.5%

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

   if -3.3999999999999999e-11 < z < 1.44999999999999997e-38

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-11} \lor \neg \left(z \leq 1.45 \cdot 10^{-38}\right):\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2150000000 \lor \neg \left(z \leq 2350000\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2150000000.0) (not (<= z 2350000.0)))
   (* (- x t) z)
   (fma (- t x) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2150000000.0) || !(z <= 2350000.0)) {
		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 <= -2150000000.0) || !(z <= 2350000.0))
		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, -2150000000.0], N[Not[LessEqual[z, 2350000.0]], $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 < -2.15e9 or 2.35e6 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

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

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

       3. mul-1-neg N/A

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

       4. mul-1-neg N/A

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

       5. *-rgt-identity N/A

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

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

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

       9. distribute-lft-neg-in N/A

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

       10. mul-1-neg N/A

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

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

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

       12. *-rgt-identity N/A

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

       13. distribute-neg-in N/A

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

       14. +-commutative N/A

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

       15. mul-1-neg N/A

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

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

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

       17. metadata-eval N/A

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

       18. *-lft-identity N/A

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

       19. mul-1-neg N/A

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

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

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

       21. *-commutative N/A

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

       22. *-commutative N/A

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

   11. Applied rewrites 83.1%

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

   if -2.15e9 < z < 2.35e6

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2150000000 \lor \neg \left(z \leq 2350000\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2150000000 \lor \neg \left(z \leq 1.45 \cdot 10^{-38}\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2150000000.0) (not (<= z 1.45e-38)))
   (* (- x t) z)
   (* (- t x) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2150000000.0) || !(z <= 1.45e-38)) {
		tmp = (x - t) * z;
	} else {
		tmp = (t - x) * 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 <= (-2150000000.0d0)) .or. (.not. (z <= 1.45d-38))) then
        tmp = (x - t) * z
    else
        tmp = (t - x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2150000000.0) || !(z <= 1.45e-38)) {
		tmp = (x - t) * z;
	} else {
		tmp = (t - x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2150000000.0) or not (z <= 1.45e-38):
		tmp = (x - t) * z
	else:
		tmp = (t - x) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2150000000.0) || !(z <= 1.45e-38))
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = Float64(Float64(t - x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2150000000.0) || ~((z <= 1.45e-38)))
		tmp = (x - t) * z;
	else
		tmp = (t - x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2150000000.0], N[Not[LessEqual[z, 1.45e-38]], $MachinePrecision]], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e9 or 1.44999999999999997e-38 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.9%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

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

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

       3. mul-1-neg N/A

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

       4. mul-1-neg N/A

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

       5. *-rgt-identity N/A

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

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

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

       7. *-lft-identity N/A

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

       8. metadata-eval N/A

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

       9. distribute-lft-neg-in N/A

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

       10. mul-1-neg N/A

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

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

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

       12. *-rgt-identity N/A

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

       13. distribute-neg-in N/A

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

       14. +-commutative N/A

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

       15. mul-1-neg N/A

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

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

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

       17. metadata-eval N/A

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

       18. *-lft-identity N/A

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

       19. mul-1-neg N/A

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

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

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

       21. *-commutative N/A

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

       22. *-commutative N/A

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

   11. Applied rewrites 81.6%

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

   if -2.15e9 < z < 1.44999999999999997e-38

   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 62.3

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

   5. Applied rewrites 62.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2150000000 \lor \neg \left(z \leq 1.45 \cdot 10^{-38}\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000165 \lor \neg \left(y \leq 16500000000\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -0.000165) (not (<= y 16500000000.0)))
   (* (- t x) y)
   (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -0.000165) || !(y <= 16500000000.0)) {
		tmp = (t - x) * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -0.000165) || !(y <= 16500000000.0))
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -0.000165], N[Not[LessEqual[y, 16500000000.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e-4 or 1.65e10 < 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 77.0

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

   5. Applied rewrites 77.0%

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

   if -1.65e-4 < y < 1.65e10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-in N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

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

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.0%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000165 \lor \neg \left(y \leq 16500000000\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-57} \lor \neg \left(x \leq 1.1 \cdot 10^{-124}\right):\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.3e-57) (not (<= x 1.1e-124))) (fma x z x) (* (- t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.3e-57) || !(x <= 1.1e-124)) {
		tmp = fma(x, z, x);
	} else {
		tmp = -t * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.3e-57) || !(x <= 1.1e-124))
		tmp = fma(x, z, x);
	else
		tmp = Float64(Float64(-t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.3e-57], N[Not[LessEqual[x, 1.1e-124]], $MachinePrecision]], N[(x * z + x), $MachinePrecision], N[((-t) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.30000000000000022e-57 or 1.0999999999999999e-124 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-in N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

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

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.1%

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

   if -4.30000000000000022e-57 < x < 1.0999999999999999e-124

   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. +-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. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.6%

      \[\leadsto \left(-t\right) \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-57} \lor \neg \left(x \leq 1.1 \cdot 10^{-124}\right):\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+53} \lor \neg \left(t \leq 6.4 \cdot 10^{+80}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.1e+53) (not (<= t 6.4e+80))) (* t y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.1e+53) || !(t <= 6.4e+80)) {
		tmp = t * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.1e+53) || !(t <= 6.4e+80))
		tmp = Float64(t * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.1e+53], N[Not[LessEqual[t, 6.4e+80]], $MachinePrecision]], N[(t * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000002e53 or 6.39999999999999979e80 < t

   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 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.4%

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

   if -2.1000000000000002e53 < t < 6.39999999999999979e80

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-in N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

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

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 51.7%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+53} \lor \neg \left(t \leq 6.4 \cdot 10^{+80}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2200000000 \lor \neg \left(z \leq 660000\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2200000000.0) (not (<= z 660000.0))) (* x z) (* t y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2200000000.0) || !(z <= 660000.0)) {
		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 <= (-2200000000.0d0)) .or. (.not. (z <= 660000.0d0))) 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 <= -2200000000.0) || !(z <= 660000.0)) {
		tmp = x * z;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2200000000.0) or not (z <= 660000.0):
		tmp = x * z
	else:
		tmp = t * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2200000000.0) || !(z <= 660000.0))
		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 <= -2200000000.0) || ~((z <= 660000.0)))
		tmp = x * z;
	else
		tmp = t * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2200000000.0], N[Not[LessEqual[z, 660000.0]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(t * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e9 or 6.6e5 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-in N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

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

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.6%

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

   if -2.2e9 < z < 6.6e5

   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 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.5%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2200000000 \lor \neg \left(z \leq 660000\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.9% 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 42.5

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

5. Applied rewrites 42.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.0%

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

Developer Target 1: 96.2% 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 2024352 
(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))))