Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Percentage Accurate: 92.6% → 97.9%

Time: 8.5s

Alternatives: 9

Speedup: 1.1×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 - x)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 - x)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 97.9

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

3. Applied rewrites 97.9%

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

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot \left(z - x\right)\\ t_2 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+273}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y t) (- z x))) (t_2 (+ x (/ (* y (- z x)) t))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1e+273) (+ x (/ (* y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) * (z - x);
	double t_2 = x + ((y * (z - x)) / t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+273) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / t) * (z - x);
	double t_2 = x + ((y * (z - x)) / t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 1e+273) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / t) * (z - x)
	t_2 = x + ((y * (z - x)) / t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 1e+273:
		tmp = x + ((y * z) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) * Float64(z - x))
	t_2 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+273)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / t) * (z - x);
	t_2 = x + ((y * (z - x)) / t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 1e+273)
		tmp = x + ((y * z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+273], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0 or 9.99999999999999945e272 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 58.4

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

   8. Applied rewrites 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.3

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

   10. Applied rewrites 61.3%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 9.99999999999999945e272

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-88}:\\ \;\;\;\;x - \frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -1.26e-76) t_1 (if (<= z 4.7e-88) (- x (* (/ y t) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (z <= -1.26e-76) {
		tmp = t_1;
	} else if (z <= 4.7e-88) {
		tmp = x - ((y / t) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (z <= -1.26e-76)
		tmp = t_1;
	elseif (z <= 4.7e-88)
		tmp = Float64(x - Float64(Float64(y / t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1.26e-76], t$95$1, If[LessEqual[z, 4.7e-88], N[(x - N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.26e-76 or 4.7e-88 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 77.0

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

   6. Applied rewrites 77.0%

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

   if -1.26e-76 < z < 4.7e-88

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 65.0

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

   8. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. remove-double-neg N/A

         \[\leadsto x - \frac{y}{t} \cdot x \]

      9. lower-*.f64 65.0

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

   10. Applied rewrites 65.0%

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

Alternative 4: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -1.26e-76) t_1 (if (<= z 4.7e-88) (* x (- 1.0 (/ y t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (z <= -1.26e-76) {
		tmp = t_1;
	} else if (z <= 4.7e-88) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (z <= -1.26e-76)
		tmp = t_1;
	elseif (z <= 4.7e-88)
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1.26e-76], t$95$1, If[LessEqual[z, 4.7e-88], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.26e-76 or 4.7e-88 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 77.0

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

   6. Applied rewrites 77.0%

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

   if -1.26e-76 < z < 4.7e-88

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 65.0

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

   8. Applied rewrites 65.0%

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

Alternative 5: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-153}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.5e-60)
   (fma (/ z t) y x)
   (if (<= t 7e-153) (/ (* y (- z x)) t) (fma (/ y t) z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.5e-60) {
		tmp = fma((z / t), y, x);
	} else if (t <= 7e-153) {
		tmp = (y * (z - x)) / t;
	} else {
		tmp = fma((y / t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.5e-60)
		tmp = fma(Float64(z / t), y, x);
	elseif (t <= 7e-153)
		tmp = Float64(Float64(y * Float64(z - x)) / t);
	else
		tmp = fma(Float64(y / t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.5e-60], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 7e-153], N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.50000000000000009e-60

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if -1.50000000000000009e-60 < t < 6.99999999999999961e-153

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if 6.99999999999999961e-153 < t

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 77.0

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

   6. Applied rewrites 77.0%

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

Alternative 6: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot \left(z - x\right)\\ t_2 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y t) (- z x))) (t_2 (+ x (/ (* y (- z x)) t))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 1e+273) (fma (/ y t) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) * (z - x);
	double t_2 = x + ((y * (z - x)) / t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+273) {
		tmp = fma((y / t), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) * Float64(z - x))
	t_2 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+273)
		tmp = fma(Float64(y / t), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+273], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0 or 9.99999999999999945e272 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 58.4

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

   8. Applied rewrites 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.3

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

   10. Applied rewrites 61.3%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 9.99999999999999945e272

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 77.0

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

   6. Applied rewrites 77.0%

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

Alternative 7: 77.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 73.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. lower-fma.f64 77.0

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

6. Applied rewrites 77.0%

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

Alternative 8: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 860000000:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y t) z)))
   (if (<= y -9.5e-65) t_1 (if (<= y 860000000.0) (* x 1.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) * z;
	double tmp;
	if (y <= -9.5e-65) {
		tmp = t_1;
	} else if (y <= 860000000.0) {
		tmp = x * 1.0;
	} 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 = (y / t) * z
    if (y <= (-9.5d-65)) then
        tmp = t_1
    else if (y <= 860000000.0d0) then
        tmp = x * 1.0d0
    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 = (y / t) * z;
	double tmp;
	if (y <= -9.5e-65) {
		tmp = t_1;
	} else if (y <= 860000000.0) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / t) * z
	tmp = 0
	if y <= -9.5e-65:
		tmp = t_1
	elif y <= 860000000.0:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) * z)
	tmp = 0.0
	if (y <= -9.5e-65)
		tmp = t_1;
	elseif (y <= 860000000.0)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / t) * z;
	tmp = 0.0;
	if (y <= -9.5e-65)
		tmp = t_1;
	elseif (y <= 860000000.0)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -9.5e-65], t$95$1, If[LessEqual[y, 860000000.0], N[(x * 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5000000000000004e-65 or 8.6e8 < y

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.9

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

   4. Applied rewrites 37.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 41.1

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

   6. Applied rewrites 41.1%

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

   if -9.5000000000000004e-65 < y < 8.6e8

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 65.0

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

   8. Applied rewrites 65.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 38.4%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z, double t) {
	return x * 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * 1.0;
}

Python

def code(x, y, z, t):
	return x * 1.0

Julia

function code(x, y, z, t)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 97.9

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

3. Applied rewrites 97.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip-rev N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift--.f64 N/A

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

   7. lower--.f64 N/A

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

   8. distribute-neg-frac N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-negate-rev N/A

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

   16. lower--.f64 92.6

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

5. Applied rewrites 92.6%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-/.f64 65.0

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

8. Applied rewrites 65.0%

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

9. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
10. Step-by-step derivation

11. Applied rewrites 38.4%

    \[\leadsto x \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64
  (+ x (/ (* y (- z x)) t)))