Average Error: 10.4 → 3.2
Time: 7.3s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z} \]
↓
\[\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a} \]
(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))
double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}
↓
double code(double x, double y, double z, double t, double a) {
	return (x / (t - (a * z))) - (y / ((t / z) - a));
}

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 * z))
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 / (t - (a * z))) - (y / ((t / z) - a))
end function

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

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

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

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

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

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

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

Original10.4
Target1.9
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

  1. Initial program 10.4

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

  2. Taylor expanded in x around 0 10.4

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

  3. Applied *-un-lft-identity_binary6410.4

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

  4. Applied times-frac_binary648.2

    \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{1} \cdot \frac{z}{t - a \cdot z}} \]

  5. Simplified8.2

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

  6. Simplified8.2

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

  7. Taylor expanded in y around 0 10.4

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

  8. Simplified3.2

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

  9. Final simplification3.2

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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