Average Error: 1.4 → 1.3
Time: 4.9s
Precision: binary64
\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[x + \frac{y}{\frac{a - z}{t - z}} \]
(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 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 - 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 - 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 - 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 - 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 - 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 - 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 - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Original1.4
Target1.3
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  3. Taylor expanded in y around 0 10.3

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

  4. Simplified2.9

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

  5. Applied egg-rr1.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2022150 
(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)))))