Average Error: 1.2 → 1.1
Time: 5.7s
Precision: binary64
\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[x + \frac{y}{\frac{a}{t - z} - \frac{z}{t - z}} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+ x (/ y (- (/ a (- t z)) (/ z (- t z))))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return x + (y / ((a / (t - z)) - (z / (t - z))));
}

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y * ((z - t) / (z - a)))
end function
↓
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y / ((a / (t - z)) - (z / (t - z))))
end function

public static double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
public static double code(double x, double y, double z, double t, double a) {
	return x + (y / ((a / (t - z)) - (z / (t - z))));
}

def code(x, y, z, t, a):
	return x + (y * ((z - t) / (z - a)))
↓
def code(x, y, z, t, a):
	return x + (y / ((a / (t - z)) - (z / (t - z))))

function code(x, y, z, t, a)
	return Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
end
↓
function code(x, y, z, t, a)
	return Float64(x + Float64(y / Float64(Float64(a / Float64(t - z)) - Float64(z / Float64(t - z)))))
end

function tmp = code(x, y, z, t, a)
	tmp = x + (y * ((z - t) / (z - a)));
end
↓
function tmp = code(x, y, z, t, a)
	tmp = x + (y / ((a / (t - z)) - (z / (t - z))));
end

code[x_, y_, z_, t_, a_] := N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := N[(x + N[(y / N[(N[(a / N[(t - z), $MachinePrecision]), $MachinePrecision] - N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x + y \cdot \frac{z - t}{z - a}
↓
x + \frac{y}{\frac{a}{t - z} - \frac{z}{t - z}}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.2
Target1.1
Herbie1.1
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Initial program 1.2

    \[x + y \cdot \frac{z - t}{z - a} \]

  2. Simplified1.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a - z}, x\right)} \]

  3. Taylor expanded in y around 0 10.4

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

  4. Simplified3.0

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

  5. Applied egg-rr1.1

    \[\leadsto x + \color{blue}{\frac{y}{\frac{a - z}{t - z}}} \]

  6. Taylor expanded in a around 0 1.1

    \[\leadsto x + \frac{y}{\color{blue}{\frac{a}{t - z} - \frac{z}{t - z}}} \]

  7. Final simplification1.1

    \[\leadsto x + \frac{y}{\frac{a}{t - z} - \frac{z}{t - z}} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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