Average Error: 10.9 → 1.3
Time: 4.5s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
↓
\[\left(\frac{y}{\frac{a - t}{z}} + x\right) - y \cdot \frac{t}{a - t} \]
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (- (+ (/ y (/ (- a t) z)) x) (* y (/ t (- a t)))))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (a - t));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((y / ((a - t) / z)) + x) - (y * (t / (a - t)));
}

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)) / (a - t))
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 = ((y / ((a - t) / z)) + x) - (y * (t / (a - t)))
end function

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

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

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

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

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

x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\left(\frac{y}{\frac{a - t}{z}} + x\right) - y \cdot \frac{t}{a - t}

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

Original10.9
Target1.3
Herbie1.3
\[x + \frac{y}{\frac{a - t}{z - t}} \]

Derivation

  1. Initial program 10.9

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

  2. Simplified1.4

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

  3. Taylor expanded in y around 0 10.9

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

  4. Simplified3.2

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

  5. Applied egg-rr5.1

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

  6. Applied egg-rr1.3

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

  7. Final simplification1.3

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

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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