(/ z0 (* z1 z2))

Percentage Accurate: 92.5% → 99.7%

Time: 13.9s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\frac{z0}{z1 \cdot z2} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (/ z0 (* z1 z2)))

C

double code(double z0, double z1, double z2) {
	return z0 / (z1 * z2);
}

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(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = z0 / (z1 * z2)
end function

Java

public static double code(double z0, double z1, double z2) {
	return z0 / (z1 * z2);
}

Python

def code(z0, z1, z2):
	return z0 / (z1 * z2)

Julia

function code(z0, z1, z2)
	return Float64(z0 / Float64(z1 * z2))
end

MATLAB

function tmp = code(z0, z1, z2)
	tmp = z0 / (z1 * z2);
end

Wolfram

code[z0_, z1_, z2_] := N[(z0 / N[(z1 * z2), $MachinePrecision]), $MachinePrecision]

Initial Program: 92.5% accurate, 1.0× speedup

Math

\[\frac{z0}{z1 \cdot z2} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (/ z0 (* z1 z2)))

C

double code(double z0, double z1, double z2) {
	return z0 / (z1 * z2);
}

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(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = z0 / (z1 * z2)
end function

Java

public static double code(double z0, double z1, double z2) {
	return z0 / (z1 * z2);
}

Python

def code(z0, z1, z2):
	return z0 / (z1 * z2)

Julia

function code(z0, z1, z2)
	return Float64(z0 / Float64(z1 * z2))
end

MATLAB

function tmp = code(z0, z1, z2)
	tmp = z0 / (z1 * z2);
end

Wolfram

code[z0_, z1_, z2_] := N[(z0 / N[(z1 * z2), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z1\right|, \left|z2\right|\right)\\ t_1 := \mathsf{max}\left(\left|z1\right|, \left|z2\right|\right)\\ \mathsf{copysign}\left(1, z0\right) \cdot \left(\mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z2\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|z0\right|}{t\_0 \cdot t\_1} \leq 10000000000000000000:\\ \;\;\;\;\frac{\frac{\left|z0\right|}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|z0\right|}{t\_1}}{t\_0}\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (let* ((t_0 (fmin (fabs z1) (fabs z2)))
       (t_1 (fmax (fabs z1) (fabs z2))))
  (*
   (copysign 1 z0)
   (*
    (copysign 1 z1)
    (*
     (copysign 1 z2)
     (if (<= (/ (fabs z0) (* t_0 t_1)) 10000000000000000000)
       (/ (/ (fabs z0) t_0) t_1)
       (/ (/ (fabs z0) t_1) t_0)))))))

C

double code(double z0, double z1, double z2) {
	double t_0 = fmin(fabs(z1), fabs(z2));
	double t_1 = fmax(fabs(z1), fabs(z2));
	double tmp;
	if ((fabs(z0) / (t_0 * t_1)) <= 1e+19) {
		tmp = (fabs(z0) / t_0) / t_1;
	} else {
		tmp = (fabs(z0) / t_1) / t_0;
	}
	return copysign(1.0, z0) * (copysign(1.0, z1) * (copysign(1.0, z2) * tmp));
}

Java

public static double code(double z0, double z1, double z2) {
	double t_0 = fmin(Math.abs(z1), Math.abs(z2));
	double t_1 = fmax(Math.abs(z1), Math.abs(z2));
	double tmp;
	if ((Math.abs(z0) / (t_0 * t_1)) <= 1e+19) {
		tmp = (Math.abs(z0) / t_0) / t_1;
	} else {
		tmp = (Math.abs(z0) / t_1) / t_0;
	}
	return Math.copySign(1.0, z0) * (Math.copySign(1.0, z1) * (Math.copySign(1.0, z2) * tmp));
}

Python

def code(z0, z1, z2):
	t_0 = fmin(math.fabs(z1), math.fabs(z2))
	t_1 = fmax(math.fabs(z1), math.fabs(z2))
	tmp = 0
	if (math.fabs(z0) / (t_0 * t_1)) <= 1e+19:
		tmp = (math.fabs(z0) / t_0) / t_1
	else:
		tmp = (math.fabs(z0) / t_1) / t_0
	return math.copysign(1.0, z0) * (math.copysign(1.0, z1) * (math.copysign(1.0, z2) * tmp))

Julia

function code(z0, z1, z2)
	t_0 = fmin(abs(z1), abs(z2))
	t_1 = fmax(abs(z1), abs(z2))
	tmp = 0.0
	if (Float64(abs(z0) / Float64(t_0 * t_1)) <= 1e+19)
		tmp = Float64(Float64(abs(z0) / t_0) / t_1);
	else
		tmp = Float64(Float64(abs(z0) / t_1) / t_0);
	end
	return Float64(copysign(1.0, z0) * Float64(copysign(1.0, z1) * Float64(copysign(1.0, z2) * tmp)))
end

MATLAB

function tmp_2 = code(z0, z1, z2)
	t_0 = min(abs(z1), abs(z2));
	t_1 = max(abs(z1), abs(z2));
	tmp = 0.0;
	if ((abs(z0) / (t_0 * t_1)) <= 1e+19)
		tmp = (abs(z0) / t_0) / t_1;
	else
		tmp = (abs(z0) / t_1) / t_0;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * ((sign(z1) * abs(1.0)) * ((sign(z2) * abs(1.0)) * tmp));
end

Wolfram

code[z0_, z1_, z2_] := Block[{t$95$0 = N[Min[N[Abs[z1], $MachinePrecision], N[Abs[z2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z1], $MachinePrecision], N[Abs[z2], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Abs[z0], $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 10000000000000000000], N[(N[(N[Abs[z0], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Abs[z0], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z0 (*.f64 z1 z2)) < 1e19

   1. Initial program 92.5%

      \[\frac{z0}{z1 \cdot z2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0}{z1 \cdot z2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0}{\color{blue}{z1 \cdot z2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z0}{z1}}{z2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z0}{z1}}{z2}} \]

      5. lower-/.f64 91.6%

         \[\leadsto \frac{\color{blue}{\frac{z0}{z1}}}{z2} \]

   3. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\frac{z0}{z1}}{z2}} \]

   if 1e19 < (/.f64 z0 (*.f64 z1 z2))

   1. Initial program 92.5%

      \[\frac{z0}{z1 \cdot z2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0}{z1 \cdot z2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0}{\color{blue}{z1 \cdot z2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{z0}{\color{blue}{z2 \cdot z1}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z0}{z2}}{z1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z0}{z2}}{z1}} \]

      6. lower-/.f64 92.1%

         \[\leadsto \frac{\color{blue}{\frac{z0}{z2}}}{z1} \]

   3. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{\frac{z0}{z2}}{z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{z0}{\left|z1\right|}}{\left|z2\right|}\\ t_1 := \left|z1\right| \cdot \left|z2\right|\\ \mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z2\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq \frac{5357543035931337}{10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000000000003960719125422883827062840959584985546704194967116721787948758551386272267278602882264876081416647209031203419106557526049419390978660438178426771560410745940876447333535260291112887354734608898565252528592034690824272687386622186778733613155375371021108230826846322688:\\ \;\;\;\;\frac{z0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (let* ((t_0 (/ (/ z0 (fabs z1)) (fabs z2)))
       (t_1 (* (fabs z1) (fabs z2))))
  (*
   (copysign 1 z1)
   (*
    (copysign 1 z2)
    (if (<=
         t_1
         5357543035931337/10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376)
      t_0
      (if (<=
           t_1
           50000000000000003960719125422883827062840959584985546704194967116721787948758551386272267278602882264876081416647209031203419106557526049419390978660438178426771560410745940876447333535260291112887354734608898565252528592034690824272687386622186778733613155375371021108230826846322688)
        (/ z0 t_1)
        t_0))))))

C

double code(double z0, double z1, double z2) {
	double t_0 = (z0 / fabs(z1)) / fabs(z2);
	double t_1 = fabs(z1) * fabs(z2);
	double tmp;
	if (t_1 <= 5e-286) {
		tmp = t_0;
	} else if (t_1 <= 5e+283) {
		tmp = z0 / t_1;
	} else {
		tmp = t_0;
	}
	return copysign(1.0, z1) * (copysign(1.0, z2) * tmp);
}

Java

public static double code(double z0, double z1, double z2) {
	double t_0 = (z0 / Math.abs(z1)) / Math.abs(z2);
	double t_1 = Math.abs(z1) * Math.abs(z2);
	double tmp;
	if (t_1 <= 5e-286) {
		tmp = t_0;
	} else if (t_1 <= 5e+283) {
		tmp = z0 / t_1;
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z2) * tmp);
}

Python

def code(z0, z1, z2):
	t_0 = (z0 / math.fabs(z1)) / math.fabs(z2)
	t_1 = math.fabs(z1) * math.fabs(z2)
	tmp = 0
	if t_1 <= 5e-286:
		tmp = t_0
	elif t_1 <= 5e+283:
		tmp = z0 / t_1
	else:
		tmp = t_0
	return math.copysign(1.0, z1) * (math.copysign(1.0, z2) * tmp)

Julia

function code(z0, z1, z2)
	t_0 = Float64(Float64(z0 / abs(z1)) / abs(z2))
	t_1 = Float64(abs(z1) * abs(z2))
	tmp = 0.0
	if (t_1 <= 5e-286)
		tmp = t_0;
	elseif (t_1 <= 5e+283)
		tmp = Float64(z0 / t_1);
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z2) * tmp))
end

MATLAB

function tmp_2 = code(z0, z1, z2)
	t_0 = (z0 / abs(z1)) / abs(z2);
	t_1 = abs(z1) * abs(z2);
	tmp = 0.0;
	if (t_1 <= 5e-286)
		tmp = t_0;
	elseif (t_1 <= 5e+283)
		tmp = z0 / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = (sign(z1) * abs(1.0)) * ((sign(z2) * abs(1.0)) * tmp);
end

Wolfram

code[z0_, z1_, z2_] := Block[{t$95$0 = N[(N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision] / N[Abs[z2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z1], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 5357543035931337/10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376], t$95$0, If[LessEqual[t$95$1, 50000000000000003960719125422883827062840959584985546704194967116721787948758551386272267278602882264876081416647209031203419106557526049419390978660438178426771560410745940876447333535260291112887354734608898565252528592034690824272687386622186778733613155375371021108230826846322688], N[(z0 / t$95$1), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z1 z2) < 5.0000000000000004e-286 or 5.0000000000000004e283 < (*.f64 z1 z2)

   1. Initial program 92.5%

      \[\frac{z0}{z1 \cdot z2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0}{z1 \cdot z2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0}{\color{blue}{z1 \cdot z2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z0}{z1}}{z2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z0}{z1}}{z2}} \]

      5. lower-/.f64 91.6%

         \[\leadsto \frac{\color{blue}{\frac{z0}{z1}}}{z2} \]

   3. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\frac{z0}{z1}}{z2}} \]

   if 5.0000000000000004e-286 < (*.f64 z1 z2) < 5.0000000000000004e283

   1. Initial program 92.5%

      \[\frac{z0}{z1 \cdot z2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0 z1 z2)
  :name "(/ z0 (* z1 z2))"
  :precision binary64
  (/ z0 (* z1 z2)))