Quotient of products

Percentage Accurate: 86.8% → 96.4%

Time: 4.3s

Alternatives: 5

Speedup: 0.2×

Specification

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a2}{b1 \cdot b2} \end{array} \]

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

real(8) function code(a1, a2, b1, b2)
    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: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a2}{b1 \cdot b2} \end{array} \]

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

real(8) function code(a1, a2, b1, b2)
    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: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-322}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -2e+282)
     (* (/ a2 b1) (/ a1 b2))
     (if (<= t_0 -1e-322)
       t_0
       (if (<= t_0 0.0)
         (* a1 (/ (/ a2 b1) b2))
         (if (<= t_0 5e+281) t_0 (/ (/ a2 b2) (/ b1 a1))))))))

C

double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+282) {
		tmp = (a2 / b1) * (a1 / b2);
	} else if (t_0 <= -1e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+281) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) / (b1 / a1);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a1 * a2) / (b1 * b2)
    if (t_0 <= (-2d+282)) then
        tmp = (a2 / b1) * (a1 / b2)
    else if (t_0 <= (-1d-322)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = a1 * ((a2 / b1) / b2)
    else if (t_0 <= 5d+281) then
        tmp = t_0
    else
        tmp = (a2 / b2) / (b1 / a1)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+282) {
		tmp = (a2 / b1) * (a1 / b2);
	} else if (t_0 <= -1e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+281) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) / (b1 / a1);
	}
	return tmp;
}

Python

def code(a1, a2, b1, b2):
	t_0 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_0 <= -2e+282:
		tmp = (a2 / b1) * (a1 / b2)
	elif t_0 <= -1e-322:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = a1 * ((a2 / b1) / b2)
	elif t_0 <= 5e+281:
		tmp = t_0
	else:
		tmp = (a2 / b2) / (b1 / a1)
	return tmp

Julia

function code(a1, a2, b1, b2)
	t_0 = Float64(Float64(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -2e+282)
		tmp = Float64(Float64(a2 / b1) * Float64(a1 / b2));
	elseif (t_0 <= -1e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(a1 * Float64(Float64(a2 / b1) / b2));
	elseif (t_0 <= 5e+281)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 / b2) / Float64(b1 / a1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, b1, b2)
	t_0 = (a1 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -2e+282)
		tmp = (a2 / b1) * (a1 / b2);
	elseif (t_0 <= -1e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = a1 * ((a2 / b1) / b2);
	elseif (t_0 <= 5e+281)
		tmp = t_0;
	else
		tmp = (a2 / b2) / (b1 / a1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+282], N[(N[(a2 / b1), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-322], t$95$0, If[LessEqual[t$95$0, 0.0], N[(a1 * N[(N[(a2 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+281], t$95$0, N[(N[(a2 / b2), $MachinePrecision] / N[(b1 / a1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.00000000000000007e282

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. times-frac 88.2%

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

   3. Applied egg-rr 88.2%

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

   if -2.00000000000000007e282 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.88131e-323 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.00000000000000016e281

   1. Initial program 98.7%

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

   if -9.88131e-323 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 70.5%

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

      1. times-frac 96.5%

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

      2. *-commutative 96.5%

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

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
   4. Step-by-step derivation

      1. clear-num 96.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{b2}{a2}}} \cdot \frac{a1}{b1} \]

      2. frac-times 96.6%

         \[\leadsto \color{blue}{\frac{1 \cdot a1}{\frac{b2}{a2} \cdot b1}} \]

      3. *-un-lft-identity 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in a1 around 0 70.5%

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]
   7. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. associate-*r/ 90.0%

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

      3. associate-/l/ 96.7%

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

   8. Simplified 96.7%

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

   if 5.00000000000000016e281 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 66.5%

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

      1. times-frac 98.0%

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

      2. *-commutative 98.0%

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

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
   4. Step-by-step derivation

      1. clear-num 98.0%

         \[\leadsto \frac{a2}{b2} \cdot \color{blue}{\frac{1}{\frac{b1}{a1}}} \]

      2. un-div-inv 98.1%

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

   5. Applied egg-rr 98.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

Alternative 2: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-322}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -2e+282)
     (* (/ a2 b1) (/ a1 b2))
     (if (<= t_0 -1e-322)
       t_0
       (if (<= t_0 0.0)
         (* a1 (/ (/ a2 b1) b2))
         (if (<= t_0 5e+281) t_0 (* (/ a2 b2) (/ a1 b1))))))))

C

double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+282) {
		tmp = (a2 / b1) * (a1 / b2);
	} else if (t_0 <= -1e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+281) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a1 * a2) / (b1 * b2)
    if (t_0 <= (-2d+282)) then
        tmp = (a2 / b1) * (a1 / b2)
    else if (t_0 <= (-1d-322)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = a1 * ((a2 / b1) / b2)
    else if (t_0 <= 5d+281) then
        tmp = t_0
    else
        tmp = (a2 / b2) * (a1 / b1)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+282) {
		tmp = (a2 / b1) * (a1 / b2);
	} else if (t_0 <= -1e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+281) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	return tmp;
}

Python

def code(a1, a2, b1, b2):
	t_0 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_0 <= -2e+282:
		tmp = (a2 / b1) * (a1 / b2)
	elif t_0 <= -1e-322:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = a1 * ((a2 / b1) / b2)
	elif t_0 <= 5e+281:
		tmp = t_0
	else:
		tmp = (a2 / b2) * (a1 / b1)
	return tmp

Julia

function code(a1, a2, b1, b2)
	t_0 = Float64(Float64(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -2e+282)
		tmp = Float64(Float64(a2 / b1) * Float64(a1 / b2));
	elseif (t_0 <= -1e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(a1 * Float64(Float64(a2 / b1) / b2));
	elseif (t_0 <= 5e+281)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 / b2) * Float64(a1 / b1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, b1, b2)
	t_0 = (a1 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -2e+282)
		tmp = (a2 / b1) * (a1 / b2);
	elseif (t_0 <= -1e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = a1 * ((a2 / b1) / b2);
	elseif (t_0 <= 5e+281)
		tmp = t_0;
	else
		tmp = (a2 / b2) * (a1 / b1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+282], N[(N[(a2 / b1), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-322], t$95$0, If[LessEqual[t$95$0, 0.0], N[(a1 * N[(N[(a2 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+281], t$95$0, N[(N[(a2 / b2), $MachinePrecision] * N[(a1 / b1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.00000000000000007e282

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. times-frac 88.2%

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

   3. Applied egg-rr 88.2%

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

   if -2.00000000000000007e282 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.88131e-323 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.00000000000000016e281

   1. Initial program 98.7%

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

   if -9.88131e-323 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 70.5%

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

      1. times-frac 96.5%

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

      2. *-commutative 96.5%

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

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
   4. Step-by-step derivation

      1. clear-num 96.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{b2}{a2}}} \cdot \frac{a1}{b1} \]

      2. frac-times 96.6%

         \[\leadsto \color{blue}{\frac{1 \cdot a1}{\frac{b2}{a2} \cdot b1}} \]

      3. *-un-lft-identity 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in a1 around 0 70.5%

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]
   7. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. associate-*r/ 90.0%

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

      3. associate-/l/ 96.7%

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

   8. Simplified 96.7%

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

   if 5.00000000000000016e281 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 66.5%

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

      1. times-frac 98.0%

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

      2. *-commutative 98.0%

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

   3. Simplified 98.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternative 3: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq -6 \cdot 10^{-107}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= a2 -6e-107) (* (/ a2 b1) (/ a1 b2)) (* a1 (/ (/ a2 b1) b2))))

C

double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (a2 <= -6e-107) {
		tmp = (a2 / b1) * (a1 / b2);
	} else {
		tmp = a1 * ((a2 / b1) / b2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    real(8) :: tmp
    if (a2 <= (-6d-107)) then
        tmp = (a2 / b1) * (a1 / b2)
    else
        tmp = a1 * ((a2 / b1) / b2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (a2 <= -6e-107) {
		tmp = (a2 / b1) * (a1 / b2);
	} else {
		tmp = a1 * ((a2 / b1) / b2);
	}
	return tmp;
}

Python

def code(a1, a2, b1, b2):
	tmp = 0
	if a2 <= -6e-107:
		tmp = (a2 / b1) * (a1 / b2)
	else:
		tmp = a1 * ((a2 / b1) / b2)
	return tmp

Julia

function code(a1, a2, b1, b2)
	tmp = 0.0
	if (a2 <= -6e-107)
		tmp = Float64(Float64(a2 / b1) * Float64(a1 / b2));
	else
		tmp = Float64(a1 * Float64(Float64(a2 / b1) / b2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, b1, b2)
	tmp = 0.0;
	if (a2 <= -6e-107)
		tmp = (a2 / b1) * (a1 / b2);
	else
		tmp = a1 * ((a2 / b1) / b2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, b1_, b2_] := If[LessEqual[a2, -6e-107], N[(N[(a2 / b1), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision], N[(a1 * N[(N[(a2 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < -5.9999999999999994e-107

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. times-frac 89.9%

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

   3. Applied egg-rr 89.9%

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

   if -5.9999999999999994e-107 < a2

   1. Initial program 80.7%

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

      1. times-frac 84.1%

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

      2. *-commutative 84.1%

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

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
   4. Step-by-step derivation

      1. clear-num 84.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{b2}{a2}}} \cdot \frac{a1}{b1} \]

      2. frac-times 91.0%

         \[\leadsto \color{blue}{\frac{1 \cdot a1}{\frac{b2}{a2} \cdot b1}} \]

      3. *-un-lft-identity 91.0%

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

   5. Applied egg-rr 91.0%

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

   6. Taylor expanded in a1 around 0 80.7%

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]
   7. Step-by-step derivation

      1. *-commutative 80.7%

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

      2. associate-*r/ 86.3%

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

      3. associate-/l/ 88.1%

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

   8. Simplified 88.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq -6 \cdot 10^{-107}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array} \]

Alternative 4: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= a2 -3.2e-105) (* (/ a2 b1) (/ a1 b2)) (/ a1 (* b1 (/ b2 a2)))))

C

double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (a2 <= -3.2e-105) {
		tmp = (a2 / b1) * (a1 / b2);
	} else {
		tmp = a1 / (b1 * (b2 / a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    real(8) :: tmp
    if (a2 <= (-3.2d-105)) then
        tmp = (a2 / b1) * (a1 / b2)
    else
        tmp = a1 / (b1 * (b2 / a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (a2 <= -3.2e-105) {
		tmp = (a2 / b1) * (a1 / b2);
	} else {
		tmp = a1 / (b1 * (b2 / a2));
	}
	return tmp;
}

Python

def code(a1, a2, b1, b2):
	tmp = 0
	if a2 <= -3.2e-105:
		tmp = (a2 / b1) * (a1 / b2)
	else:
		tmp = a1 / (b1 * (b2 / a2))
	return tmp

Julia

function code(a1, a2, b1, b2)
	tmp = 0.0
	if (a2 <= -3.2e-105)
		tmp = Float64(Float64(a2 / b1) * Float64(a1 / b2));
	else
		tmp = Float64(a1 / Float64(b1 * Float64(b2 / a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, b1, b2)
	tmp = 0.0;
	if (a2 <= -3.2e-105)
		tmp = (a2 / b1) * (a1 / b2);
	else
		tmp = a1 / (b1 * (b2 / a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, b1_, b2_] := If[LessEqual[a2, -3.2e-105], N[(N[(a2 / b1), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision], N[(a1 / N[(b1 * N[(b2 / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < -3.19999999999999981e-105

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. times-frac 89.9%

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

   3. Applied egg-rr 89.9%

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

   if -3.19999999999999981e-105 < a2

   1. Initial program 80.7%

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

      1. times-frac 84.1%

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

      2. *-commutative 84.1%

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

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
   4. Step-by-step derivation

      1. clear-num 84.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{b2}{a2}}} \cdot \frac{a1}{b1} \]

      2. frac-times 91.0%

         \[\leadsto \color{blue}{\frac{1 \cdot a1}{\frac{b2}{a2} \cdot b1}} \]

      3. *-un-lft-identity 91.0%

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

   5. Applied egg-rr 91.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ a1 \cdot \frac{\frac{a2}{b1}}{b2} \end{array} \]

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

real(8) function code(a1, a2, b1, b2)
    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(a1 * Float64(Float64(a2 / b1) / b2))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. times-frac 83.4%

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

   2. *-commutative 83.4%

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

3. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
4. Step-by-step derivation

   1. clear-num 83.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{b2}{a2}}} \cdot \frac{a1}{b1} \]

   2. frac-times 86.6%

      \[\leadsto \color{blue}{\frac{1 \cdot a1}{\frac{b2}{a2} \cdot b1}} \]

   3. *-un-lft-identity 86.6%

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

5. Applied egg-rr 86.6%

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

6. Taylor expanded in a1 around 0 80.4%

   \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]
7. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. associate-*r/ 84.9%

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

   3. associate-/l/ 86.1%

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

8. Simplified 86.1%

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

9. Final simplification 86.1%

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

Developer target: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a1}{b1} \cdot \frac{a2}{b2} \end{array} \]

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

real(8) function code(a1, a2, b1, b2)
    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 2023299 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

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