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

Average Error: 7.6 → 2.1

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot z - x\\ \mathbf{if}\;t \leq 4.21312888900773 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t_1}, \frac{x}{x + 1}\right) - \frac{x}{\left(x + 1\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{y}{t} - \frac{x}{t_1}\right)}{x + 1}\\ \end{array} \]

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
 (let* ((t_1 (- (* t z) x)))
   (if (<= t 4.21312888900773e-16)
     (-
      (fma (/ y (+ x 1.0)) (/ z t_1) (/ x (+ x 1.0)))
      (/ x (* (+ x 1.0) t_1)))
     (/ (+ x (- (/ y t) (/ x t_1))) (+ 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) {
	double t_1 = (t * z) - x;
	double tmp;
	if (t <= 4.21312888900773e-16) {
		tmp = fma((y / (x + 1.0)), (z / t_1), (x / (x + 1.0))) - (x / ((x + 1.0) * t_1));
	} else {
		tmp = (x + ((y / t) - (x / t_1))) / (x + 1.0);
	}
	return tmp;
}

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)
	t_1 = Float64(Float64(t * z) - x)
	tmp = 0.0
	if (t <= 4.21312888900773e-16)
		tmp = Float64(fma(Float64(y / Float64(x + 1.0)), Float64(z / t_1), Float64(x / Float64(x + 1.0))) - Float64(x / Float64(Float64(x + 1.0) * t_1)));
	else
		tmp = Float64(Float64(x + Float64(Float64(y / t) - Float64(x / t_1))) / Float64(x + 1.0));
	end
	return tmp
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_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t, 4.21312888900773e-16], N[(N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y / t), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.6
Target	0.3
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if t < 4.2131288890077302e-16

   1. Initial program 7.0

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

   2. Taylor expanded in y around 0 7.0

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

   3. Applied egg-rr 1.6

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

   if 4.2131288890077302e-16 < t

   1. Initial program 9.2

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

   2. Taylor expanded in y around 0 9.2

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

   3. Simplified 5.4

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

   4. Taylor expanded in z around inf 3.3

      \[\leadsto \frac{x + \left(\color{blue}{\frac{y}{t}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.21312888900773 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t \cdot z - x}, \frac{x}{x + 1}\right) - \frac{x}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{y}{t} - \frac{x}{t \cdot z - x}\right)}{x + 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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)))