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

Percentage Accurate: 90.2% → 96.4%

Time: 21.7s

Precision: binary64

Cost: 8644

• Unsound rule application detected in e-graph. Results may not be sound.

• 47.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := a + b \cdot c\\ \mathbf{if}\;\left(z \cdot t + x \cdot y\right) - \left(c \cdot t_1\right) \cdot i \leq \infty:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - t_1 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(c \cdot \left(t_1 \cdot \left(-i\right)\right)\right)\\ \end{array} \]

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 (+ a (* b c))))
   (if (<= (- (+ (* z t) (* x y)) (* (* c t_1) i)) INFINITY)
     (* 2.0 (- (fma x y (* z t)) (* t_1 (* c i))))
     (* 2.0 (* c (* t_1 (- i)))))))

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 = a + (b * c);
	double tmp;
	if ((((z * t) + (x * y)) - ((c * t_1) * i)) <= ((double) INFINITY)) {
		tmp = 2.0 * (fma(x, y, (z * t)) - (t_1 * (c * i)));
	} else {
		tmp = 2.0 * (c * (t_1 * -i));
	}
	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 = Float64(a + Float64(b * c))
	tmp = 0.0
	if (Float64(Float64(Float64(z * t) + Float64(x * y)) - Float64(Float64(c * t_1) * i)) <= Inf)
		tmp = Float64(2.0 * Float64(fma(x, y, Float64(z * t)) - Float64(t_1 * Float64(c * i))));
	else
		tmp = Float64(2.0 * Float64(c * Float64(t_1 * Float64(-i))));
	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[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(c * t$95$1), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 * N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c * N[(t$95$1 * (-i)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.2%
Target	94.2%
Herbie	96.4%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)) < +inf.0

   1. Initial program 91.6%

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

   2. Simplified 97.9%

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

      [Start] 91.6	\[ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
      associate-*l* [=>] 97.9	\[ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right) \]
      fma-def [=>] 97.9	\[ 2 \cdot \left(\color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right)} - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

   if +inf.0 < (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))

   1. Initial program 0.0%

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

   2. Taylor expanded in i around inf 61.5%

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(c \cdot \left(i \cdot \left(c \cdot b + a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

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

Alternatives

Alternative 1
Accuracy	94.9%
Cost	2504

\[\begin{array}{l} t_1 := a + b \cdot c\\ t_2 := \left(c \cdot t_1\right) \cdot i\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+304}:\\ \;\;\;\;2 \cdot \left(c \cdot \left(t_1 \cdot \left(-i\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq 10^{+307}:\\ \;\;\;\;\left(\left(z \cdot t + x \cdot y\right) - t_2\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(t_1 \cdot i\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	67.4%
Cost	1364

\[\begin{array}{l} t_1 := 2 \cdot \left(z \cdot t - c \cdot \left(a \cdot i\right)\right)\\ t_2 := -2 \cdot \left(b \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-75}:\\ \;\;\;\;\left(z \cdot t + x \cdot y\right) \cdot 2\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	73.0%
Cost	1361

\[\begin{array}{l} t_1 := 2 \cdot \left(c \cdot \left(\left(a + b \cdot c\right) \cdot \left(-i\right)\right)\right)\\ \mathbf{if}\;c \leq -7.3 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\left(z \cdot t + x \cdot y\right) \cdot 2\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+65} \lor \neg \left(c \leq 3.65 \cdot 10^{+137}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	78.9%
Cost	1356

\[\begin{array}{l} t_1 := c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\\ t_2 := 2 \cdot \left(x \cdot y - t_1\right)\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\left(z \cdot t + x \cdot y\right) \cdot 2\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t - t_1\right)\\ \end{array} \]

Alternative 5
Accuracy	73.2%
Cost	1298

\[\begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-21} \lor \neg \left(c \leq 2.5 \cdot 10^{-64}\right) \land \left(c \leq 2.05 \cdot 10^{+67} \lor \neg \left(c \leq 2.4 \cdot 10^{+80}\right)\right):\\ \;\;\;\;2 \cdot \left(c \cdot \left(\left(a + b \cdot c\right) \cdot \left(-i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t + x \cdot y\right) \cdot 2\\ \end{array} \]

Alternative 6
Accuracy	79.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{-65} \lor \neg \left(c \leq 6.2 \cdot 10^{-76}\right):\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t + x \cdot y\right) \cdot 2\\ \end{array} \]

Alternative 7
Accuracy	85.8%
Cost	1224

\[\begin{array}{l} t_1 := c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot \left(x \cdot y - t_1\right)\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(\left(z \cdot t + x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t - t_1\right)\\ \end{array} \]

Alternative 8
Accuracy	49.0%
Cost	1104

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot y\right)\\ t_2 := -2 \cdot \left(b \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{if}\;c \leq -2.9 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-171}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	65.9%
Cost	1104

\[\begin{array}{l} t_1 := \left(z \cdot t + x \cdot y\right) \cdot 2\\ t_2 := -2 \cdot \left(b \cdot \left(c \cdot \left(c \cdot i\right)\right)\right)\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot \left(i \cdot \left(c \cdot \left(-a\right)\right)\right)\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	38.2%
Cost	844

\[\begin{array}{l} t_1 := a \cdot \left(\left(c \cdot i\right) \cdot -2\right)\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	39.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-22} \lor \neg \left(x \leq 3.9 \cdot 10^{-75}\right):\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \end{array} \]

Alternative 12
Accuracy	28.7%
Cost	320

\[2 \cdot \left(z \cdot t\right) \]

Reproduce

herbie shell --seed 2023161 
(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))))