Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, B

Average Error: 0.0 → 9.1

Time: 3.8s

Precision: binary64

Cost: 7108

Math

\[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, 0.125 \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 10^{+24}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + \left(y \cdot z\right) \cdot -0.5\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y z) -5e+121)
   (fma y (* z -0.5) (* 0.125 x))
   (if (<= (* y z) 1e+24) (+ (* 0.125 x) t) (+ (* 0.125 x) (* (* y z) -0.5)))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * z) <= -5e+121) {
		tmp = fma(y, (z * -0.5), (0.125 * x));
	} else if ((y * z) <= 1e+24) {
		tmp = (0.125 * x) + t;
	} else {
		tmp = (0.125 * x) + ((y * z) * -0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * z) <= -5e+121)
		tmp = fma(y, Float64(z * -0.5), Float64(0.125 * x));
	elseif (Float64(y * z) <= 1e+24)
		tmp = Float64(Float64(0.125 * x) + t);
	else
		tmp = Float64(Float64(0.125 * x) + Float64(Float64(y * z) * -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(y * z), $MachinePrecision], -5e+121], N[(y * N[(z * -0.5), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 1e+24], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Target

Original	0.0
Target	0.0
Herbie	9.1

\[\left(\frac{x}{8} + t\right) - \frac{z}{2} \cdot y \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5.00000000000000007e121

   1. Initial program 0.0

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in t around 0 10.0

      \[\leadsto \color{blue}{0.125 \cdot x - 0.5 \cdot \left(y \cdot z\right)} \]

   3. Simplified 9.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot -0.5, 0.125 \cdot x\right)} \]

   if -5.00000000000000007e121 < (*.f64 y z) < 9.9999999999999998e23

   1. Initial program 0.0

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in y around 0 7.7

      \[\leadsto \color{blue}{0.125 \cdot x + t} \]

   if 9.9999999999999998e23 < (*.f64 y z)

   1. Initial program 0.1

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. Taylor expanded in t around 0 13.5

      \[\leadsto \color{blue}{0.125 \cdot x - 0.5 \cdot \left(y \cdot z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, 0.125 \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 10^{+24}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + \left(y \cdot z\right) \cdot -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	9.1
Cost	1096

\[\begin{array}{l} t_1 := 0.125 \cdot x + \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \cdot z \leq 10^{+24}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	9.5
Cost	968

\[\begin{array}{l} t_1 := t + \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -2.0915082795430662 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \cdot z \leq 1.0761491174801814 \cdot 10^{+124}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.8
Cost	840

\[\begin{array}{l} t_1 := \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -7.20243854477268 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \cdot z \leq 1.0761491174801814 \cdot 10^{+124}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	28.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7296732402236914 \cdot 10^{+81}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq -3.353267083893106 \cdot 10^{-6}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;t \leq -1.7350310241546463 \cdot 10^{-56}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 4.111892382606937 \cdot 10^{+94}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 5
Error	28.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7350310241546463 \cdot 10^{-56}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 4.111892382606937 \cdot 10^{+94}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 6
Error	39.9
Cost	64

\[t \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (+ (/ x 8.0) t) (* (/ z 2.0) y))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))