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

Average Error: 6.3 → 0.3

Time: 10.6s

Precision: binary64

Math

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(x, y, z \cdot t\right)\\ t_2 := 2 \cdot \mathsf{fma}\left(c, -\mathsf{fma}\left(c \cdot i, b, a \cdot i\right), t_1\right)\\ t_3 := c \cdot \left(a + b \cdot c\right)\\ \mathbf{if}\;t_3 \leq -3.251941869890704 \cdot 10^{+300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 5.423452089292153 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \left(t_1 - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* z t)))
        (t_2 (* 2.0 (fma c (- (fma (* c i) b (* a i))) t_1)))
        (t_3 (* c (+ a (* b c)))))
   (if (<= t_3 -3.251941869890704e+300)
     t_2
     (if (<= t_3 5.423452089292153e+198)
       (* 2.0 (- t_1 (* i (* c (fma b c a)))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (z * t));
	double t_2 = 2.0 * fma(c, -fma((c * i), b, (a * i)), t_1);
	double t_3 = c * (a + (b * c));
	double tmp;
	if (t_3 <= -3.251941869890704e+300) {
		tmp = t_2;
	} else if (t_3 <= 5.423452089292153e+198) {
		tmp = 2.0 * (t_1 - (i * (c * fma(b, c, a))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

↓

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(z * t))
	t_2 = Float64(2.0 * fma(c, Float64(-fma(Float64(c * i), b, Float64(a * i))), t_1))
	t_3 = Float64(c * Float64(a + Float64(b * c)))
	tmp = 0.0
	if (t_3 <= -3.251941869890704e+300)
		tmp = t_2;
	elseif (t_3 <= 5.423452089292153e+198)
		tmp = Float64(2.0 * Float64(t_1 - Float64(i * Float64(c * fma(b, c, a)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(c * (-N[(N[(c * i), $MachinePrecision] * b + N[(a * i), $MachinePrecision]), $MachinePrecision]) + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -3.251941869890704e+300], t$95$2, If[LessEqual[t$95$3, 5.423452089292153e+198], N[(2.0 * N[(t$95$1 - N[(i * N[(c * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Target

Original	6.3
Target	1.9
Herbie	0.3

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -3.25194186989070401e300 or 5.423452089292153e198 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 42.4

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Simplified 42.4

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]

   3. Applied egg-rr 7.2

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(c, -\mathsf{fma}\left(c, b, a\right) \cdot i, \mathsf{fma}\left(x, y, z \cdot t\right)\right)} \]

   4. Taylor expanded in c around 0 3.4

      \[\leadsto 2 \cdot \mathsf{fma}\left(c, -\color{blue}{\left(c \cdot \left(i \cdot b\right) + a \cdot i\right)}, \mathsf{fma}\left(x, y, z \cdot t\right)\right) \]

   5. Applied egg-rr 0.5

      \[\leadsto 2 \cdot \mathsf{fma}\left(c, -\color{blue}{\mathsf{fma}\left(c \cdot i, b, i \cdot a\right)}, \mathsf{fma}\left(x, y, z \cdot t\right)\right) \]

   if -3.25194186989070401e300 < (*.f64 (+.f64 a (*.f64 b c)) c) < 5.423452089292153e198

   1. Initial program 0.3

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Simplified 0.3

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot \left(a + b \cdot c\right) \leq -3.251941869890704 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(c, -\mathsf{fma}\left(c \cdot i, b, a \cdot i\right), \mathsf{fma}\left(x, y, z \cdot t\right)\right)\\ \mathbf{elif}\;c \cdot \left(a + b \cdot c\right) \leq 5.423452089292153 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(c, -\mathsf{fma}\left(c \cdot i, b, a \cdot i\right), \mathsf{fma}\left(x, y, z \cdot t\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* 2.0 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))