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

Average Error: 0.1 → 0.1

Time: 7.6s

Precision: binary64

Cost: 6848

Math

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

↓

\[0.5 \cdot \left(y \cdot \sqrt{z} + x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

↓

(FPCore (x y z) :precision binary64 (* 0.5 (+ (* y (sqrt z)) x)))

C

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

↓

double code(double x, double y, double z) {
	return 0.5 * ((y * sqrt(z)) + x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * ((y * sqrt(z)) + x)
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

↓

public static double code(double x, double y, double z) {
	return 0.5 * ((y * Math.sqrt(z)) + x);
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

↓

def code(x, y, z):
	return 0.5 * ((y * math.sqrt(z)) + x)

Julia

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

↓

function code(x, y, z)
	return Float64(0.5 * Float64(Float64(y * sqrt(z)) + x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

↓

function tmp = code(x, y, z)
	tmp = 0.5 * ((y * sqrt(z)) + x);
end

Wolfram

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

↓

code[x_, y_, z_] := N[(0.5 * N[(N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
   Proof
   (*.f64 1/2 (fma.f64 y (sqrt.f64 z) x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (fma.f64 y (sqrt.f64 z) x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 1 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (sqrt.f64 z)) x))): 1 points increase in error, 1 points decrease in error
   (*.f64 (/.f64 1 2) (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (sqrt.f64 z))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in y around 0 0.1

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

4. Final simplification 0.1

   \[\leadsto 0.5 \cdot \left(y \cdot \sqrt{z} + x\right) \]

Alternatives

Alternative 1
Error	16.2
Cost	20040

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ t_1 := 0.5 \cdot t_0\\ \mathbf{if}\;t_0 \leq -200000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	29.9
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))