Average Error: 10.4 → 0.3
Time: 7.0s
Precision: binary64
Cost: 841
\[\frac{x + y \cdot \left(z - x\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+92} \lor \neg \left(y \leq 1.05 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{\frac{z}{1 - y}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+92) (not (<= y 1.05e+15)))
   (/ y (/ z (- z x)))
   (+ y (/ x (/ z (- 1.0 y))))))
double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+92) || !(y <= 1.05e+15)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = y + (x / (z / (1.0 - 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 - x))) / 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) :: tmp
    if ((y <= (-5d+92)) .or. (.not. (y <= 1.05d+15))) then
        tmp = y / (z / (z - x))
    else
        tmp = y + (x / (z / (1.0d0 - y)))
    end if
    code = tmp
end function

public static double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}
↓
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+92) || !(y <= 1.05e+15)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = y + (x / (z / (1.0 - y)));
	}
	return tmp;
}

def code(x, y, z):
	return (x + (y * (z - x))) / z
↓
def code(x, y, z):
	tmp = 0
	if (y <= -5e+92) or not (y <= 1.05e+15):
		tmp = y / (z / (z - x))
	else:
		tmp = y + (x / (z / (1.0 - y)))
	return tmp

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end
↓
function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e+92) || !(y <= 1.05e+15))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(y + Float64(x / Float64(z / Float64(1.0 - y))));
	end
	return tmp
end

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end
↓
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5e+92) || ~((y <= 1.05e+15)))
		tmp = y / (z / (z - x));
	else
		tmp = y + (x / (z / (1.0 - y)));
	end
	tmp_2 = tmp;
end

code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]
↓
code[x_, y_, z_] := If[Or[LessEqual[y, -5e+92], N[Not[LessEqual[y, 1.05e+15]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / N[(z / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -5 \cdot 10^{+92} \lor \neg \left(y \leq 1.05 \cdot 10^{+15}\right):\\
\;\;\;\;\frac{y}{\frac{z}{z - x}}\\

\mathbf{else}:\\
\;\;\;\;y + \frac{x}{\frac{z}{1 - y}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.4
Target0.0
Herbie0.3
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.00000000000000022e92 or 1.05e15 < y

    1. Initial program 27.0

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

    2. Taylor expanded in y around inf 27.0

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

    3. Simplified0.1

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

      [Start]27.0

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

      associate-/l* [=>]0.1

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

    if -5.00000000000000022e92 < y < 1.05e15

    1. Initial program 1.1

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

    2. Taylor expanded in x around inf 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{y + \frac{x}{\frac{z}{1 - y}}} \]
      Proof

      [Start]0.4

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

      *-commutative [=>]0.4

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

      associate-/l* [=>]0.4

      \[ y + \color{blue}{\frac{x}{\frac{z}{1 + -1 \cdot y}}} \]

      mul-1-neg [=>]0.4

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

      unsub-neg [=>]0.4

      \[ y + \frac{x}{\frac{z}{\color{blue}{1 - y}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+92} \lor \neg \left(y \leq 1.05 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{\frac{z}{1 - y}}\\ \end{array} \]

Alternatives

Alternative 1
Error21.1
Cost720
\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-34}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-202}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 2
Error0.8
Cost713
\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
Alternative 3
Error0.8
Cost713
\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
Alternative 4
Error9.4
Cost712
\[\begin{array}{l} \mathbf{if}\;y \leq 0.022:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 5
Error9.1
Cost648
\[\begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+33}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 6
Error8.8
Cost320
\[y + \frac{x}{z} \]
Alternative 7
Error31.4
Cost64
\[y \]

Error

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))