Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Average Error: 0.0 → 0.0

Time: 7.3s

Precision: binary64

Cost: 640

Math

\[x + \left(y - x\right) \cdot z \]

↓

\[z \cdot y + \left(-1 + z\right) \cdot \left(-x\right) \]

FPCore

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

↓

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

C

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

↓

double code(double x, double y, double z) {
	return (z * y) + ((-1.0 + 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 = x + ((y - x) * 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 = (z * y) + (((-1.0d0) + z) * -x)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

   \[x + \left(y - x\right) \cdot z \]

2. Taylor expanded in x around -inf 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ y \cdot z + -1 \cdot \left(\left(z - 1\right) \cdot x\right) \]
   rational_best-simplify-2 [=>] 0.0	\[ \color{blue}{z \cdot y} + -1 \cdot \left(\left(z - 1\right) \cdot x\right) \]
   rational_best-simplify-44 [=>] 0.0	\[ z \cdot y + \color{blue}{\left(z - 1\right) \cdot \left(-1 \cdot x\right)} \]
   rational_best-simplify-19 [=>] 0.0	\[ z \cdot y + \color{blue}{\left(z + -1\right)} \cdot \left(-1 \cdot x\right) \]
   rational_best-simplify-1 [=>] 0.0	\[ z \cdot y + \color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot x\right) \]
   rational_best-simplify-2 [=>] 0.0	\[ z \cdot y + \left(-1 + z\right) \cdot \color{blue}{\left(x \cdot -1\right)} \]
   rational_best-simplify-12 [=>] 0.0	\[ z \cdot y + \left(-1 + z\right) \cdot \color{blue}{\left(-x\right)} \]

4. Final simplification 0.0

   \[\leadsto z \cdot y + \left(-1 + z\right) \cdot \left(-x\right) \]

Alternatives

Alternative 1
Error	16.9
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	23.7
Cost	784

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-14}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+27}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.7
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -72000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	25.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

\[x + \left(y - x\right) \cdot z \]

Alternative 7
Error	35.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023092 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))