Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 62.0% → 91.5%

Time: 4.5s

Alternatives: 8

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq \frac{8409788860576655}{5922386521532855740161817506647119732883018558947359509044845726112560091729648156474603305162988578607512400425457279991804428268870599332596921062626576000993556884845161077691136496092218188572933193945756793025561702170624}:\\ \;\;\;\;\frac{1}{\sqrt{\left(-t\right) \cdot a}} \cdot \left(\left(\left|z\right| \cdot t\_1\right) \cdot t\_2\right)\\ \mathbf{elif}\;\left|z\right| \leq 139999999999999991285167330229307404975375020111088221626015833673130848699810489861179420414813756105335038937414138448123527168:\\ \;\;\;\;\left(\frac{\left|z\right|}{\sqrt{\left|z\right| \cdot \left|z\right| - t \cdot a}} \cdot t\_2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (*
     (copysign 1 z)
     (if (<=
          (fabs z)
          8409788860576655/5922386521532855740161817506647119732883018558947359509044845726112560091729648156474603305162988578607512400425457279991804428268870599332596921062626576000993556884845161077691136496092218188572933193945756793025561702170624)
       (* (/ 1 (sqrt (* (- t) a))) (* (* (fabs z) t_1) t_2))
       (if (<=
            (fabs z)
            139999999999999991285167330229307404975375020111088221626015833673130848699810489861179420414813756105335038937414138448123527168)
         (*
          (*
           (/ (fabs z) (sqrt (- (* (fabs z) (fabs z)) (* t a))))
           t_2)
          t_1)
         (* 1 (* t_1 t_2)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.42e-210) {
		tmp = (1.0 / sqrt((-t * a))) * ((fabs(z) * t_1) * t_2);
	} else if (fabs(z) <= 1.4e+128) {
		tmp = ((fabs(z) / sqrt(((fabs(z) * fabs(z)) - (t * a)))) * t_2) * t_1;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.42e-210) {
		tmp = (1.0 / Math.sqrt((-t * a))) * ((Math.abs(z) * t_1) * t_2);
	} else if (Math.abs(z) <= 1.4e+128) {
		tmp = ((Math.abs(z) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (t * a)))) * t_2) * t_1;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.42e-210:
		tmp = (1.0 / math.sqrt((-t * a))) * ((math.fabs(z) * t_1) * t_2)
	elif math.fabs(z) <= 1.4e+128:
		tmp = ((math.fabs(z) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (t * a)))) * t_2) * t_1
	else:
		tmp = 1.0 * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.42e-210)
		tmp = Float64(Float64(1.0 / sqrt(Float64(Float64(-t) * a))) * Float64(Float64(abs(z) * t_1) * t_2));
	elseif (abs(z) <= 1.4e+128)
		tmp = Float64(Float64(Float64(abs(z) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(t * a)))) * t_2) * t_1);
	else
		tmp = Float64(1.0 * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.42e-210)
		tmp = (1.0 / sqrt((-t * a))) * ((abs(z) * t_1) * t_2);
	elseif (abs(z) <= 1.4e+128)
		tmp = ((abs(z) / sqrt(((abs(z) * abs(z)) - (t * a)))) * t_2) * t_1;
	else
		tmp = 1.0 * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 8409788860576655/5922386521532855740161817506647119732883018558947359509044845726112560091729648156474603305162988578607512400425457279991804428268870599332596921062626576000993556884845161077691136496092218188572933193945756793025561702170624], N[(N[(1 / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z], $MachinePrecision], 139999999999999991285167330229307404975375020111088221626015833673130848699810489861179420414813756105335038937414138448123527168], N[(N[(N[(N[Abs[z], $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], N[(1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.42e-210

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.8%

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

   3. Applied rewrites 59.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{y}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \cdot \left(z \cdot x\right) \]

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 32.2%

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 32.5%

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

   if 1.42e-210 < z < 1.3999999999999999e128

   1. Initial program 62.0%

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

   3. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt-neg-out N/A

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

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

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 64.0%

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

   if 1.3999999999999999e128 < z

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 43.6%

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

Alternative 2: 91.4% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 139999999999999991285167330229307404975375020111088221626015833673130848699810489861179420414813756105335038937414138448123527168:\\ \;\;\;\;\frac{\left|z\right|}{\sqrt{\left|z\right| \cdot \left|z\right| - a \cdot t}} \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (*
 (copysign 1 z)
 (if (<=
      (fabs z)
      139999999999999991285167330229307404975375020111088221626015833673130848699810489861179420414813756105335038937414138448123527168)
   (* (/ (fabs z) (sqrt (- (* (fabs z) (fabs z)) (* a t)))) (* y x))
   (* 1 (* y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (fabs(z) <= 1.4e+128) {
		tmp = (fabs(z) / sqrt(((fabs(z) * fabs(z)) - (a * t)))) * (y * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return copysign(1.0, z) * tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.abs(z) <= 1.4e+128) {
		tmp = (Math.abs(z) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (a * t)))) * (y * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return Math.copySign(1.0, z) * tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.fabs(z) <= 1.4e+128:
		tmp = (math.fabs(z) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (a * t)))) * (y * x)
	else:
		tmp = 1.0 * (y * x)
	return math.copysign(1.0, z) * tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (abs(z) <= 1.4e+128)
		tmp = Float64(Float64(abs(z) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(a * t)))) * Float64(y * x));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(copysign(1.0, z) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (abs(z) <= 1.4e+128)
		tmp = (abs(z) / sqrt(((abs(z) * abs(z)) - (a * t)))) * (y * x);
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = (sign(z) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 139999999999999991285167330229307404975375020111088221626015833673130848699810489861179420414813756105335038937414138448123527168], N[(N[(N[Abs[z], $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(1 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.3999999999999999e128

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   if 1.3999999999999999e128 < z

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 43.6%

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

Alternative 3: 90.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq \frac{8409788860576655}{5922386521532855740161817506647119732883018558947359509044845726112560091729648156474603305162988578607512400425457279991804428268870599332596921062626576000993556884845161077691136496092218188572933193945756793025561702170624}:\\ \;\;\;\;\frac{1}{\sqrt{\left(-t\right) \cdot a}} \cdot \left(\left(\left|z\right| \cdot t\_1\right) \cdot t\_2\right)\\ \mathbf{elif}\;\left|z\right| \leq 124999999999999997902848:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{\left|z\right| \cdot \left|z\right| - t \cdot a}} \cdot \left|z\right|\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (*
     (copysign 1 z)
     (if (<=
          (fabs z)
          8409788860576655/5922386521532855740161817506647119732883018558947359509044845726112560091729648156474603305162988578607512400425457279991804428268870599332596921062626576000993556884845161077691136496092218188572933193945756793025561702170624)
       (* (/ 1 (sqrt (* (- t) a))) (* (* (fabs z) t_1) t_2))
       (if (<= (fabs z) 124999999999999997902848)
         (*
          (*
           (/ t_2 (sqrt (- (* (fabs z) (fabs z)) (* t a))))
           (fabs z))
          t_1)
         (* 1 (* t_1 t_2)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.42e-210) {
		tmp = (1.0 / sqrt((-t * a))) * ((fabs(z) * t_1) * t_2);
	} else if (fabs(z) <= 1.25e+23) {
		tmp = ((t_2 / sqrt(((fabs(z) * fabs(z)) - (t * a)))) * fabs(z)) * t_1;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.42e-210) {
		tmp = (1.0 / Math.sqrt((-t * a))) * ((Math.abs(z) * t_1) * t_2);
	} else if (Math.abs(z) <= 1.25e+23) {
		tmp = ((t_2 / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (t * a)))) * Math.abs(z)) * t_1;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.42e-210:
		tmp = (1.0 / math.sqrt((-t * a))) * ((math.fabs(z) * t_1) * t_2)
	elif math.fabs(z) <= 1.25e+23:
		tmp = ((t_2 / math.sqrt(((math.fabs(z) * math.fabs(z)) - (t * a)))) * math.fabs(z)) * t_1
	else:
		tmp = 1.0 * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.42e-210)
		tmp = Float64(Float64(1.0 / sqrt(Float64(Float64(-t) * a))) * Float64(Float64(abs(z) * t_1) * t_2));
	elseif (abs(z) <= 1.25e+23)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(t * a)))) * abs(z)) * t_1);
	else
		tmp = Float64(1.0 * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.42e-210)
		tmp = (1.0 / sqrt((-t * a))) * ((abs(z) * t_1) * t_2);
	elseif (abs(z) <= 1.25e+23)
		tmp = ((t_2 / sqrt(((abs(z) * abs(z)) - (t * a)))) * abs(z)) * t_1;
	else
		tmp = 1.0 * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 8409788860576655/5922386521532855740161817506647119732883018558947359509044845726112560091729648156474603305162988578607512400425457279991804428268870599332596921062626576000993556884845161077691136496092218188572933193945756793025561702170624], N[(N[(1 / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z], $MachinePrecision], 124999999999999997902848], N[(N[(N[(t$95$2 / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.42e-210

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.8%

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

   3. Applied rewrites 59.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{y}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \cdot \left(z \cdot x\right) \]

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 32.2%

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 32.5%

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

   if 1.42e-210 < z < 1.25e23

   1. Initial program 62.0%

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

   3. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\sqrt{z \cdot z - a \cdot t}\right)}^{\color{blue}{\left(\frac{-1}{2} + \frac{-1}{2}\right)}}\right) \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\sqrt{z \cdot z - a \cdot t}\right)}^{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{-1}{2}\right)}\right) \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\sqrt{z \cdot z - a \cdot t}\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \cdot y \]

      9. pow-add N/A

         \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot \color{blue}{\left({\left(\sqrt{z \cdot z - a \cdot t}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\sqrt{z \cdot z - a \cdot t}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \cdot y \]

      10. unpow-prod-down N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - a \cdot t} \cdot \sqrt{z \cdot z - a \cdot t}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\sqrt{z \cdot z - a \cdot t} \cdot \sqrt{z \cdot z - a \cdot t}\right)}^{\color{blue}{\frac{-1}{2}}}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\sqrt{z \cdot z - a \cdot t} \cdot \sqrt{z \cdot z - a \cdot t}\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y \]

      13. sqr-neg-rev N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)}\right) \cdot y \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}}\right) \cdot y \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \cdot y \]

      16. unpow-prod-down N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \cdot y \]

      17. pow-add N/A

          \[\leadsto \left(\left(\left(-x\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - a \cdot t}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \cdot y \]

   5. Applied rewrites 61.4%

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

   if 1.25e23 < z

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 43.6%

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

Alternative 4: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq \frac{633237988016573}{904625697166532776746648320380374280103671755200316906558262375061821325312}:\\ \;\;\;\;\frac{t\_1 \cdot \left(t\_2 \cdot \left|z\right|\right)}{\sqrt{-a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (*
     (copysign 1 z)
     (if (<=
          (fabs z)
          633237988016573/904625697166532776746648320380374280103671755200316906558262375061821325312)
       (/ (* t_1 (* t_2 (fabs z))) (sqrt (- (* a t))))
       (* 1 (* t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 7e-61) {
		tmp = (t_1 * (t_2 * fabs(z))) / sqrt(-(a * t));
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 7e-61) {
		tmp = (t_1 * (t_2 * Math.abs(z))) / Math.sqrt(-(a * t));
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 7e-61:
		tmp = (t_1 * (t_2 * math.fabs(z))) / math.sqrt(-(a * t))
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 7e-61)
		tmp = Float64(Float64(t_1 * Float64(t_2 * abs(z))) / sqrt(Float64(-Float64(a * t))));
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 7e-61)
		tmp = (t_1 * (t_2 * abs(z))) / sqrt(-(a * t));
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 633237988016573/904625697166532776746648320380374280103671755200316906558262375061821325312], N[(N[(t$95$1 * N[(t$95$2 * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(1 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 7.0000000000000006e-61

   1. Initial program 62.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 32.5%

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

   4. Applied rewrites 32.5%

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

   if 7.0000000000000006e-61 < z

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 43.6%

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

Alternative 5: 75.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq \frac{1824208758453333}{388129523075177233787244872115625638814221504279174152784763009506512738171594221582719602207161619487621932674282768301542895011028703597861071818760295284801113744005212476387566321407899611206315749798429117187723211713454014464}:\\ \;\;\;\;\frac{\left|z\right| \cdot t\_1}{-\left|z\right|} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (*
     (copysign 1 z)
     (if (<=
          (fabs z)
          1824208758453333/388129523075177233787244872115625638814221504279174152784763009506512738171594221582719602207161619487621932674282768301542895011028703597861071818760295284801113744005212476387566321407899611206315749798429117187723211713454014464)
       (* (/ (* (fabs z) t_1) (- (fabs z))) t_2)
       (* 1 (* t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 4.7e-216) {
		tmp = ((fabs(z) * t_1) / -fabs(z)) * t_2;
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 4.7e-216) {
		tmp = ((Math.abs(z) * t_1) / -Math.abs(z)) * t_2;
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 4.7e-216:
		tmp = ((math.fabs(z) * t_1) / -math.fabs(z)) * t_2
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 4.7e-216)
		tmp = Float64(Float64(Float64(abs(z) * t_1) / Float64(-abs(z))) * t_2);
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 4.7e-216)
		tmp = ((abs(z) * t_1) / -abs(z)) * t_2;
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1824208758453333/388129523075177233787244872115625638814221504279174152784763009506512738171594221582719602207161619487621932674282768301542895011028703597861071818760295284801113744005212476387566321407899611206315749798429117187723211713454014464], N[(N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] / (-N[Abs[z], $MachinePrecision])), $MachinePrecision] * t$95$2), $MachinePrecision], N[(1 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.7e-216

   1. Initial program 62.0%

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

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 42.7%

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

   4. Applied rewrites 42.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 41.1%

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

   if 4.7e-216 < z

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 43.6%

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

Alternative 6: 74.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq \frac{-5503834670046917}{20769187434139310514121985316880384}:\\ \;\;\;\;\frac{t\_1}{\left|z\right|} \cdot \left(\left|z\right| \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (*
     (copysign 1 z)
     (if (<=
          (* t a)
          -5503834670046917/20769187434139310514121985316880384)
       (* (/ t_1 (fabs z)) (* (fabs z) t_2))
       (* 1 (* t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if ((t * a) <= -2.65e-19) {
		tmp = (t_1 / fabs(z)) * (fabs(z) * t_2);
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if ((t * a) <= -2.65e-19) {
		tmp = (t_1 / Math.abs(z)) * (Math.abs(z) * t_2);
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if (t * a) <= -2.65e-19:
		tmp = (t_1 / math.fabs(z)) * (math.fabs(z) * t_2)
	else:
		tmp = 1.0 * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(t * a) <= -2.65e-19)
		tmp = Float64(Float64(t_1 / abs(z)) * Float64(abs(z) * t_2));
	else
		tmp = Float64(1.0 * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if ((t * a) <= -2.65e-19)
		tmp = (t_1 / abs(z)) * (abs(z) * t_2);
	else
		tmp = 1.0 * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t * a), $MachinePrecision], -5503834670046917/20769187434139310514121985316880384], N[(N[(t$95$1 / N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t a) < -2.6499999999999999e-19

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.8%

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

   3. Applied rewrites 59.8%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 37.5%

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

   6. Applied rewrites 37.5%

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

   if -2.6499999999999999e-19 < (*.f64 t a)

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-pow2 N/A

         \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. unpow-prod-down N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      14. pow-add N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

      17. metadata-eval N/A

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

   3. Applied rewrites 64.1%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 43.6%

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

Alternative 7: 73.3% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, z\right) \cdot \left(1 \cdot \left(y \cdot x\right)\right) \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (* (copysign 1 z) (* 1 (* y x))))

C

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

Java

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

Python

def code(x, y, z, t, a):
	return math.copysign(1.0, z) * (1.0 * (y * x))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(1 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. inv-pow N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-fabs-rev N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. sqrt-pow2 N/A

      \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}\right)}^{\left(\frac{-1}{2}\right)}} \]

   10. sqr-neg-rev N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}} \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

   13. unpow-prod-down N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

   14. pow-add N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

   16. metadata-eval N/A

       \[\leadsto \left(\left(x \cdot y\right) \cdot z\right) \cdot {\left(\mathsf{neg}\left(\sqrt{z \cdot z - t \cdot a}\right)\right)}^{\left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

   17. metadata-eval N/A

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

3. Applied rewrites 64.1%

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

4. Taylor expanded in z around inf

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

6. Applied rewrites 43.6%

   \[\leadsto \color{blue}{1} \cdot \left(y \cdot x\right) \]
7. Add Preprocessing

Alternative 8: 43.8% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in z around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 43.8%

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

4. Applied rewrites 43.8%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(y \cdot x\right) \]

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

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 43.8%

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

6. Applied rewrites 43.8%

   \[\leadsto \left(-y\right) \cdot \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64
  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))