Average Error: 6.1 → 1.9
Time: 2.8s
Precision: binary64
Cost: 1360
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -1e+201)
     t_0
     (if (<= (* x y) -5e-44)
       t_1
       (if (<= (* x y) 5e-233)
         t_0
         (if (<= (* x y) 5e+89) t_1 (/ x (/ z y))))))))
double code(double x, double y, double z) {
	return (x * y) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1e+201) {
		tmp = t_0;
	} else if ((x * y) <= -5e-44) {
		tmp = t_1;
	} else if ((x * y) <= 5e-233) {
		tmp = t_0;
	} else if ((x * y) <= 5e+89) {
		tmp = t_1;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / z
end function
↓
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (x / z)
    t_1 = (x * y) / z
    if ((x * y) <= (-1d+201)) then
        tmp = t_0
    else if ((x * y) <= (-5d-44)) then
        tmp = t_1
    else if ((x * y) <= 5d-233) then
        tmp = t_0
    else if ((x * y) <= 5d+89) then
        tmp = t_1
    else
        tmp = x / (z / y)
    end if
    code = tmp
end function

public static double code(double x, double y, double z) {
	return (x * y) / z;
}
↓
public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1e+201) {
		tmp = t_0;
	} else if ((x * y) <= -5e-44) {
		tmp = t_1;
	} else if ((x * y) <= 5e-233) {
		tmp = t_0;
	} else if ((x * y) <= 5e+89) {
		tmp = t_1;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

def code(x, y, z):
	return (x * y) / z
↓
def code(x, y, z):
	t_0 = y * (x / z)
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -1e+201:
		tmp = t_0
	elif (x * y) <= -5e-44:
		tmp = t_1
	elif (x * y) <= 5e-233:
		tmp = t_0
	elif (x * y) <= 5e+89:
		tmp = t_1
	else:
		tmp = x / (z / y)
	return tmp

function code(x, y, z)
	return Float64(Float64(x * y) / z)
end
↓
function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -1e+201)
		tmp = t_0;
	elseif (Float64(x * y) <= -5e-44)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-233)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e+89)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(z / y));
	end
	return tmp
end

function tmp = code(x, y, z)
	tmp = (x * y) / z;
end
↓
function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -1e+201)
		tmp = t_0;
	elseif ((x * y) <= -5e-44)
		tmp = t_1;
	elseif ((x * y) <= 5e-233)
		tmp = t_0;
	elseif ((x * y) <= 5e+89)
		tmp = t_1;
	else
		tmp = x / (z / y);
	end
	tmp_2 = tmp;
end

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]
↓
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+201], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -5e-44], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-233], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e+89], t$95$1, N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]]]]

\frac{x \cdot y}{z}
↓
\begin{array}{l}
t_0 := y \cdot \frac{x}{z}\\
t_1 := \frac{x \cdot y}{z}\\
\mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+201}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-44}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-233}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+89}:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target6.0
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < -1.00000000000000004e201 or -5.00000000000000039e-44 < (*.f64 x y) < 5.00000000000000012e-233

    1. Initial program 10.3

      \[\frac{x \cdot y}{z} \]

    2. Applied egg-rr2.5

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

    if -1.00000000000000004e201 < (*.f64 x y) < -5.00000000000000039e-44 or 5.00000000000000012e-233 < (*.f64 x y) < 4.99999999999999983e89

    1. Initial program 0.2

      \[\frac{x \cdot y}{z} \]

    if 4.99999999999999983e89 < (*.f64 x y)

    1. Initial program 12.0

      \[\frac{x \cdot y}{z} \]

    2. Simplified5.4

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) z)): 72 points increase in error, 50 points decrease in error

    3. Applied egg-rr5.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-233}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+89}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error6.4
Cost584
\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3184099116350476 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error6.3
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -2.3184099116350476 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
Alternative 3
Error6.1
Cost320
\[\frac{y}{\frac{z}{x}} \]

Error

Reproduce

herbie shell --seed 2022302 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))