Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.2 → 0.1

Time: 4.8s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

↓

(FPCore (x y z t)
 :precision binary64
 (+ (/ y (* (+ x 1.0) (- t (/ x z)))) (/ (- x (/ x (- (* t z) x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

↓

double code(double x, double y, double z, double t) {
	return (y / ((x + 1.0) * (t - (x / z)))) + ((x - (x / ((t * z) - x))) / (x + 1.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y / ((x + 1.0d0) * (t - (x / z)))) + ((x - (x / ((t * z) - x))) / (x + 1.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

↓

public static double code(double x, double y, double z, double t) {
	return (y / ((x + 1.0) * (t - (x / z)))) + ((x - (x / ((t * z) - x))) / (x + 1.0));
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

↓

def code(x, y, z, t):
	return (y / ((x + 1.0) * (t - (x / z)))) + ((x - (x / ((t * z) - x))) / (x + 1.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

↓

function code(x, y, z, t)
	return Float64(Float64(y / Float64(Float64(x + 1.0) * Float64(t - Float64(x / z)))) + Float64(Float64(x - Float64(x / Float64(Float64(t * z) - x))) / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

↓

function tmp = code(x, y, z, t)
	tmp = (y / ((x + 1.0) * (t - (x / z)))) + ((x - (x / ((t * z) - x))) / (x + 1.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x - N[(x / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 7.2

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

2. Taylor expanded in y around 0 7.2

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

3. Simplified 2.2

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

4. Taylor expanded in t around 0 0.3

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

5. Taylor expanded in y around 0 0.1

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

6. Simplified 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022159 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))