Quotient of products

Percentage Accurate: 85.8% → 98.7%

Time: 3.9s

Alternatives: 4

Speedup: 0.6×

• 79.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

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(a1, a2, b1, b2)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 * a2) / (b1 * b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 * a2) / Float64(b1 * b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 * a2) / (b1 * b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.8% accurate, 1.0× speedup

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

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(a1, a2, b1, b2)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 * a2) / (b1 * b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 * a2) / Float64(b1 * b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 * a2) / (b1 * b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|a1\right|, \left|a2\right|\right)\\ t_1 := \mathsf{min}\left(\left|a1\right|, \left|a2\right|\right)\\ t_2 := \frac{t\_0}{\left|b1\right|} \cdot \frac{t\_1}{\left|b2\right|}\\ t_3 := \left|b1\right| \cdot \left|b2\right|\\ t_4 := \left|b2\right| \cdot \left|b1\right|\\ \mathsf{copysign}\left(1, a1\right) \cdot \left(\mathsf{copysign}\left(1, a2\right) \cdot \left(\mathsf{copysign}\left(1, b1\right) \cdot \left(\mathsf{copysign}\left(1, b2\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 10^{-300}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5000000000:\\ \;\;\;\;\frac{t\_1}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{t\_0}{t\_4} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (fmax (fabs a1) (fabs a2)))
        (t_1 (fmin (fabs a1) (fabs a2)))
        (t_2 (* (/ t_0 (fabs b1)) (/ t_1 (fabs b2))))
        (t_3 (* (fabs b1) (fabs b2)))
        (t_4 (* (fabs b2) (fabs b1))))
   (*
    (copysign 1.0 a1)
    (*
     (copysign 1.0 a2)
     (*
      (copysign 1.0 b1)
      (*
       (copysign 1.0 b2)
       (if (<= t_3 1e-300)
         t_2
         (if (<= t_3 5000000000.0)
           (* (/ t_1 t_4) t_0)
           (if (<= t_3 2e+296) (* (/ t_0 t_4) t_1) t_2)))))))))

C

double code(double a1, double a2, double b1, double b2) {
	double t_0 = fmax(fabs(a1), fabs(a2));
	double t_1 = fmin(fabs(a1), fabs(a2));
	double t_2 = (t_0 / fabs(b1)) * (t_1 / fabs(b2));
	double t_3 = fabs(b1) * fabs(b2);
	double t_4 = fabs(b2) * fabs(b1);
	double tmp;
	if (t_3 <= 1e-300) {
		tmp = t_2;
	} else if (t_3 <= 5000000000.0) {
		tmp = (t_1 / t_4) * t_0;
	} else if (t_3 <= 2e+296) {
		tmp = (t_0 / t_4) * t_1;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, a1) * (copysign(1.0, a2) * (copysign(1.0, b1) * (copysign(1.0, b2) * tmp)));
}

Java

public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = fmax(Math.abs(a1), Math.abs(a2));
	double t_1 = fmin(Math.abs(a1), Math.abs(a2));
	double t_2 = (t_0 / Math.abs(b1)) * (t_1 / Math.abs(b2));
	double t_3 = Math.abs(b1) * Math.abs(b2);
	double t_4 = Math.abs(b2) * Math.abs(b1);
	double tmp;
	if (t_3 <= 1e-300) {
		tmp = t_2;
	} else if (t_3 <= 5000000000.0) {
		tmp = (t_1 / t_4) * t_0;
	} else if (t_3 <= 2e+296) {
		tmp = (t_0 / t_4) * t_1;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, a1) * (Math.copySign(1.0, a2) * (Math.copySign(1.0, b1) * (Math.copySign(1.0, b2) * tmp)));
}

Python

def code(a1, a2, b1, b2):
	t_0 = fmax(math.fabs(a1), math.fabs(a2))
	t_1 = fmin(math.fabs(a1), math.fabs(a2))
	t_2 = (t_0 / math.fabs(b1)) * (t_1 / math.fabs(b2))
	t_3 = math.fabs(b1) * math.fabs(b2)
	t_4 = math.fabs(b2) * math.fabs(b1)
	tmp = 0
	if t_3 <= 1e-300:
		tmp = t_2
	elif t_3 <= 5000000000.0:
		tmp = (t_1 / t_4) * t_0
	elif t_3 <= 2e+296:
		tmp = (t_0 / t_4) * t_1
	else:
		tmp = t_2
	return math.copysign(1.0, a1) * (math.copysign(1.0, a2) * (math.copysign(1.0, b1) * (math.copysign(1.0, b2) * tmp)))

Julia

function code(a1, a2, b1, b2)
	t_0 = fmax(abs(a1), abs(a2))
	t_1 = fmin(abs(a1), abs(a2))
	t_2 = Float64(Float64(t_0 / abs(b1)) * Float64(t_1 / abs(b2)))
	t_3 = Float64(abs(b1) * abs(b2))
	t_4 = Float64(abs(b2) * abs(b1))
	tmp = 0.0
	if (t_3 <= 1e-300)
		tmp = t_2;
	elseif (t_3 <= 5000000000.0)
		tmp = Float64(Float64(t_1 / t_4) * t_0);
	elseif (t_3 <= 2e+296)
		tmp = Float64(Float64(t_0 / t_4) * t_1);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, a1) * Float64(copysign(1.0, a2) * Float64(copysign(1.0, b1) * Float64(copysign(1.0, b2) * tmp))))
end

MATLAB

function tmp_2 = code(a1, a2, b1, b2)
	t_0 = max(abs(a1), abs(a2));
	t_1 = min(abs(a1), abs(a2));
	t_2 = (t_0 / abs(b1)) * (t_1 / abs(b2));
	t_3 = abs(b1) * abs(b2);
	t_4 = abs(b2) * abs(b1);
	tmp = 0.0;
	if (t_3 <= 1e-300)
		tmp = t_2;
	elseif (t_3 <= 5000000000.0)
		tmp = (t_1 / t_4) * t_0;
	elseif (t_3 <= 2e+296)
		tmp = (t_0 / t_4) * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(a1) * abs(1.0)) * ((sign(a2) * abs(1.0)) * ((sign(b1) * abs(1.0)) * ((sign(b2) * abs(1.0)) * tmp)));
end

Wolfram

code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[Max[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / N[Abs[b1], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Abs[b2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[b1], $MachinePrecision] * N[Abs[b2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[b2], $MachinePrecision] * N[Abs[b1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 1e-300], t$95$2, If[LessEqual[t$95$3, 5000000000.0], N[(N[(t$95$1 / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 2e+296], N[(N[(t$95$0 / t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$2]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < 1e-300 or 2e296 < (*.f64 b1 b2)

   1. Initial program 85.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a1}}{b1 \cdot b2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{a2 \cdot a1}{\color{blue}{b1 \cdot b2}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1}} \cdot \frac{a1}{b2} \]

      8. lower-/.f64 86.0%

         \[\leadsto \frac{a2}{b1} \cdot \color{blue}{\frac{a1}{b2}} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

   if 1e-300 < (*.f64 b1 b2) < 5e9

   1. Initial program 85.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a1 \cdot a2\right)} \cdot \frac{1}{b1 \cdot b2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a2 \cdot a1\right)} \cdot \frac{1}{b1 \cdot b2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a2 \cdot \left(a1 \cdot \frac{1}{b1 \cdot b2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1 \cdot b2}\right) \cdot a2} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1 \cdot b2}\right) \cdot a2} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

      9. lower-/.f64 86.1%

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      11. *-commutative N/A

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      12. lower-*.f64 86.1%

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

   3. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{a1}{b2 \cdot b1} \cdot a2} \]

   if 5e9 < (*.f64 b1 b2) < 2e296

   1. Initial program 85.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lower-/.f64 85.8%

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2}} \cdot a1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      8. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      9. lower-*.f64 85.8%

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

   3. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{a2}{b2 \cdot b1} \cdot a1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.0% accurate, 0.2× speedup

Math

\[\mathsf{copysign}\left(1, a1\right) \cdot \left(\mathsf{copysign}\left(1, a2\right) \cdot \left(\mathsf{copysign}\left(1, b1\right) \cdot \left(\mathsf{copysign}\left(1, b2\right) \cdot \left(\frac{\mathsf{max}\left(\left|a1\right|, \left|a2\right|\right)}{\mathsf{max}\left(\left|b1\right|, \left|b2\right|\right)} \cdot \frac{\mathsf{min}\left(\left|a1\right|, \left|a2\right|\right)}{\mathsf{min}\left(\left|b1\right|, \left|b2\right|\right)}\right)\right)\right)\right) \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (*
  (copysign 1.0 a1)
  (*
   (copysign 1.0 a2)
   (*
    (copysign 1.0 b1)
    (*
     (copysign 1.0 b2)
     (*
      (/ (fmax (fabs a1) (fabs a2)) (fmax (fabs b1) (fabs b2)))
      (/ (fmin (fabs a1) (fabs a2)) (fmin (fabs b1) (fabs b2)))))))))

C

double code(double a1, double a2, double b1, double b2) {
	return copysign(1.0, a1) * (copysign(1.0, a2) * (copysign(1.0, b1) * (copysign(1.0, b2) * ((fmax(fabs(a1), fabs(a2)) / fmax(fabs(b1), fabs(b2))) * (fmin(fabs(a1), fabs(a2)) / fmin(fabs(b1), fabs(b2)))))));
}

Java

public static double code(double a1, double a2, double b1, double b2) {
	return Math.copySign(1.0, a1) * (Math.copySign(1.0, a2) * (Math.copySign(1.0, b1) * (Math.copySign(1.0, b2) * ((fmax(Math.abs(a1), Math.abs(a2)) / fmax(Math.abs(b1), Math.abs(b2))) * (fmin(Math.abs(a1), Math.abs(a2)) / fmin(Math.abs(b1), Math.abs(b2)))))));
}

Python

def code(a1, a2, b1, b2):
	return math.copysign(1.0, a1) * (math.copysign(1.0, a2) * (math.copysign(1.0, b1) * (math.copysign(1.0, b2) * ((fmax(math.fabs(a1), math.fabs(a2)) / fmax(math.fabs(b1), math.fabs(b2))) * (fmin(math.fabs(a1), math.fabs(a2)) / fmin(math.fabs(b1), math.fabs(b2)))))))

Julia

function code(a1, a2, b1, b2)
	return Float64(copysign(1.0, a1) * Float64(copysign(1.0, a2) * Float64(copysign(1.0, b1) * Float64(copysign(1.0, b2) * Float64(Float64(fmax(abs(a1), abs(a2)) / fmax(abs(b1), abs(b2))) * Float64(fmin(abs(a1), abs(a2)) / fmin(abs(b1), abs(b2))))))))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (sign(a1) * abs(1.0)) * ((sign(a2) * abs(1.0)) * ((sign(b1) * abs(1.0)) * ((sign(b2) * abs(1.0)) * ((max(abs(a1), abs(a2)) / max(abs(b1), abs(b2))) * (min(abs(a1), abs(a2)) / min(abs(b1), abs(b2)))))));
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[Max[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision] / N[Max[N[Abs[b1], $MachinePrecision], N[Abs[b2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Min[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision] / N[Min[N[Abs[b1], $MachinePrecision], N[Abs[b2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{a1 \cdot a2}{\color{blue}{b1 \cdot b2}} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a2}{b2}} \cdot \frac{a1}{b1} \]

   8. lower-/.f64 86.6%

      \[\leadsto \frac{a2}{b2} \cdot \color{blue}{\frac{a1}{b1}} \]

3. Applied rewrites 86.6%

   \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
4. Add Preprocessing

Alternative 3: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|a1\right|, \left|a2\right|\right)\\ t_1 := \mathsf{max}\left(\left|a1\right|, \left|a2\right|\right)\\ t_2 := \left|b2\right| \cdot \left|b1\right|\\ \mathsf{copysign}\left(1, a1\right) \cdot \left(\mathsf{copysign}\left(1, a2\right) \cdot \left(\mathsf{copysign}\left(1, b1\right) \cdot \left(\mathsf{copysign}\left(1, b2\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0 \cdot t\_1}{\left|b1\right| \cdot \left|b2\right|} \leq 10^{-17}:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_2} \cdot t\_1\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (fmin (fabs a1) (fabs a2)))
        (t_1 (fmax (fabs a1) (fabs a2)))
        (t_2 (* (fabs b2) (fabs b1))))
   (*
    (copysign 1.0 a1)
    (*
     (copysign 1.0 a2)
     (*
      (copysign 1.0 b1)
      (*
       (copysign 1.0 b2)
       (if (<= (/ (* t_0 t_1) (* (fabs b1) (fabs b2))) 1e-17)
         (* (/ t_1 t_2) t_0)
         (* (/ t_0 t_2) t_1))))))))

C

double code(double a1, double a2, double b1, double b2) {
	double t_0 = fmin(fabs(a1), fabs(a2));
	double t_1 = fmax(fabs(a1), fabs(a2));
	double t_2 = fabs(b2) * fabs(b1);
	double tmp;
	if (((t_0 * t_1) / (fabs(b1) * fabs(b2))) <= 1e-17) {
		tmp = (t_1 / t_2) * t_0;
	} else {
		tmp = (t_0 / t_2) * t_1;
	}
	return copysign(1.0, a1) * (copysign(1.0, a2) * (copysign(1.0, b1) * (copysign(1.0, b2) * tmp)));
}

Java

public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = fmin(Math.abs(a1), Math.abs(a2));
	double t_1 = fmax(Math.abs(a1), Math.abs(a2));
	double t_2 = Math.abs(b2) * Math.abs(b1);
	double tmp;
	if (((t_0 * t_1) / (Math.abs(b1) * Math.abs(b2))) <= 1e-17) {
		tmp = (t_1 / t_2) * t_0;
	} else {
		tmp = (t_0 / t_2) * t_1;
	}
	return Math.copySign(1.0, a1) * (Math.copySign(1.0, a2) * (Math.copySign(1.0, b1) * (Math.copySign(1.0, b2) * tmp)));
}

Python

def code(a1, a2, b1, b2):
	t_0 = fmin(math.fabs(a1), math.fabs(a2))
	t_1 = fmax(math.fabs(a1), math.fabs(a2))
	t_2 = math.fabs(b2) * math.fabs(b1)
	tmp = 0
	if ((t_0 * t_1) / (math.fabs(b1) * math.fabs(b2))) <= 1e-17:
		tmp = (t_1 / t_2) * t_0
	else:
		tmp = (t_0 / t_2) * t_1
	return math.copysign(1.0, a1) * (math.copysign(1.0, a2) * (math.copysign(1.0, b1) * (math.copysign(1.0, b2) * tmp)))

Julia

function code(a1, a2, b1, b2)
	t_0 = fmin(abs(a1), abs(a2))
	t_1 = fmax(abs(a1), abs(a2))
	t_2 = Float64(abs(b2) * abs(b1))
	tmp = 0.0
	if (Float64(Float64(t_0 * t_1) / Float64(abs(b1) * abs(b2))) <= 1e-17)
		tmp = Float64(Float64(t_1 / t_2) * t_0);
	else
		tmp = Float64(Float64(t_0 / t_2) * t_1);
	end
	return Float64(copysign(1.0, a1) * Float64(copysign(1.0, a2) * Float64(copysign(1.0, b1) * Float64(copysign(1.0, b2) * tmp))))
end

MATLAB

function tmp_2 = code(a1, a2, b1, b2)
	t_0 = min(abs(a1), abs(a2));
	t_1 = max(abs(a1), abs(a2));
	t_2 = abs(b2) * abs(b1);
	tmp = 0.0;
	if (((t_0 * t_1) / (abs(b1) * abs(b2))) <= 1e-17)
		tmp = (t_1 / t_2) * t_0;
	else
		tmp = (t_0 / t_2) * t_1;
	end
	tmp_2 = (sign(a1) * abs(1.0)) * ((sign(a2) * abs(1.0)) * ((sign(b1) * abs(1.0)) * ((sign(b2) * abs(1.0)) * tmp)));
end

Wolfram

code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[Min[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b2], $MachinePrecision] * N[Abs[b1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(N[Abs[b1], $MachinePrecision] * N[Abs[b2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-17], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.0000000000000001e-17

   1. Initial program 85.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lower-/.f64 85.8%

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2}} \cdot a1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      8. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      9. lower-*.f64 85.8%

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

   3. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{a2}{b2 \cdot b1} \cdot a1} \]

   if 1.0000000000000001e-17 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 85.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a1 \cdot a2\right)} \cdot \frac{1}{b1 \cdot b2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a2 \cdot a1\right)} \cdot \frac{1}{b1 \cdot b2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a2 \cdot \left(a1 \cdot \frac{1}{b1 \cdot b2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1 \cdot b2}\right) \cdot a2} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1 \cdot b2}\right) \cdot a2} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

      9. lower-/.f64 86.1%

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      11. *-commutative N/A

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      12. lower-*.f64 86.1%

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

   3. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{a1}{b2 \cdot b1} \cdot a2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\frac{\mathsf{min}\left(a1, a2\right)}{b2 \cdot b1} \cdot \mathsf{max}\left(a1, a2\right) \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (* (/ (fmin a1 a2) (* b2 b1)) (fmax a1 a2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (fmin(a1, a2) / (b2 * b1)) * fmax(a1, a2);
}

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(a1, a2, b1, b2)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (fmin(a1, a2) / (b2 * b1)) * fmax(a1, a2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (fmin(a1, a2) / (b2 * b1)) * fmax(a1, a2);
}

Python

def code(a1, a2, b1, b2):
	return (fmin(a1, a2) / (b2 * b1)) * fmax(a1, a2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(fmin(a1, a2) / Float64(b2 * b1)) * fmax(a1, a2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (min(a1, a2) / (b2 * b1)) * max(a1, a2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(N[Min[a1, a2], $MachinePrecision] / N[(b2 * b1), $MachinePrecision]), $MachinePrecision] * N[Max[a1, a2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right)} \cdot \frac{1}{b1 \cdot b2} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(a2 \cdot a1\right)} \cdot \frac{1}{b1 \cdot b2} \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{a2 \cdot \left(a1 \cdot \frac{1}{b1 \cdot b2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1 \cdot b2}\right) \cdot a2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1 \cdot b2}\right) \cdot a2} \]

   8. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

   9. lower-/.f64 86.1%

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

   11. *-commutative N/A

       \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

   12. lower-*.f64 86.1%

       \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

3. Applied rewrites 86.1%

   \[\leadsto \color{blue}{\frac{a1}{b2 \cdot b1} \cdot a2} \]
4. Add Preprocessing

Developer Target 1: 86.6% accurate, 1.0× speedup

Math

\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (* (/ a1 b1) (/ a2 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 / b1) * (a2 / b2);
}

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(a1, a2, b1, b2)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 / b1) * (a2 / b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 / b1) * (a2 / b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 / b1) * (a2 / b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 / b1) * Float64(a2 / b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 / b1) * (a2 / b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025191 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :alt
  (! :herbie-platform c (* (/ a1 b1) (/ a2 b2)))

  (/ (* a1 a2) (* b1 b2)))