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

Average Error: 6.7 → 1.6

Time: 27.9s

Precision: binary64

Cost: 21128

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\right)\\ t_2 := c \cdot \left(a + b \cdot c\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+180}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - t_2 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* 2.0 (fma z t (fma (fma b c a) (* c (- i)) (* x y)))))
        (t_2 (* c (+ a (* b c)))))
   (if (<= t_2 -5e+238)
     t_1
     (if (<= t_2 5e+180) (* 2.0 (- (+ (* x y) (* z t)) (* t_2 i))) t_1))))

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 = 2.0 * fma(z, t, fma(fma(b, c, a), (c * -i), (x * y)));
	double t_2 = c * (a + (b * c));
	double tmp;
	if (t_2 <= -5e+238) {
		tmp = t_1;
	} else if (t_2 <= 5e+180) {
		tmp = 2.0 * (((x * y) + (z * t)) - (t_2 * i));
	} else {
		tmp = t_1;
	}
	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(2.0 * fma(z, t, fma(fma(b, c, a), Float64(c * Float64(-i)), Float64(x * y))))
	t_2 = Float64(c * Float64(a + Float64(b * c)))
	tmp = 0.0
	if (t_2 <= -5e+238)
		tmp = t_1;
	elseif (t_2 <= 5e+180)
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(t_2 * i)));
	else
		tmp = t_1;
	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[(2.0 * N[(z * t + N[(N[(b * c + a), $MachinePrecision] * N[(c * (-i)), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+238], t$95$1, If[LessEqual[t$95$2, 5e+180], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	6.7
Target	2.1
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.99999999999999995e238 or 4.9999999999999996e180 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 33.2

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

   2. Simplified 6.4

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), i \cdot \left(-c\right), x \cdot y\right)\right)} \]
      Proof
      (*.f64 2 (fma.f64 z t (fma.f64 (fma.f64 b c a) (*.f64 i (neg.f64 c)) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b c) a)) (*.f64 i (neg.f64 c)) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 a (*.f64 b c))) (*.f64 i (neg.f64 c)) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (fma.f64 (+.f64 a (*.f64 b c)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 i c))) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (fma.f64 (+.f64 a (*.f64 b c)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 c i))) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 a (*.f64 b c)) (neg.f64 (*.f64 c i))) (*.f64 x y))))): 2 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (+.f64 a (*.f64 b c)) (*.f64 c i)))) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))) (*.f64 x y)))): 36 points increase in error, 16 points decrease in error
      (*.f64 2 (fma.f64 z t (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (fma.f64 z t (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 z t) (-.f64 (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) (*.f64 x y))))): 2 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (*.f64 z t) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x y) (-.f64 (*.f64 z t) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)))): 0 points increase in error, 0 points decrease in error

   if -4.99999999999999995e238 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.9999999999999996e180

   1. Initial program 0.5

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot \left(a + b \cdot c\right) \leq -5 \cdot 10^{+238}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\right)\\ \mathbf{elif}\;c \cdot \left(a + b \cdot c\right) \leq 5 \cdot 10^{+180}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.2
Cost	7876

\[\begin{array}{l} t_1 := x \cdot y + z \cdot t\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;2 \cdot \left(x \cdot y - \mathsf{fma}\left(c, b, a\right) \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;2 \cdot \left(t_1 - t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 - c \cdot \left(a \cdot i\right)\right)\\ \end{array} \]

Alternative 2
Error	9.7
Cost	2892

\[\begin{array}{l} t_1 := a + b \cdot c\\ t_2 := \left(c \cdot t_1\right) \cdot i\\ t_3 := x \cdot y + z \cdot t\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+237}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(i \cdot t_1\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left(t_3 - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;2 \cdot \left(x \cdot y - t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_3 - c \cdot \left(a \cdot i\right)\right)\\ \end{array} \]

Alternative 3
Error	3.6
Cost	2504

\[\begin{array}{l} t_1 := a + b \cdot c\\ t_2 := \left(c \cdot t_1\right) \cdot i\\ t_3 := x \cdot y + z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(i \cdot t_1\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;2 \cdot \left(t_3 - t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_3 - c \cdot \left(a \cdot i\right)\right)\\ \end{array} \]

Alternative 4
Error	22.5
Cost	1760

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot y + z \cdot t\right)\\ t_2 := -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\\ \mathbf{if}\;c \leq -8 \cdot 10^{+159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.9927650023352734 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.027715178403912 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(x \cdot y - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{elif}\;c \leq 92000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{+107}:\\ \;\;\;\;2 \cdot \left(x \cdot y - i \cdot \left(c \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	22.1
Cost	1496

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot y + z \cdot t\right)\\ t_2 := -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\\ \mathbf{if}\;c \leq -8 \cdot 10^{+159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.9927650023352734 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.027715178403912 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(x \cdot y - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	23.1
Cost	1236

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot y + z \cdot t\right)\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+155}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(i \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{elif}\;c \leq -2.25 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;c \leq 3.9927650023352734 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.027715178403912 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(x \cdot y - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(c \cdot i\right) \cdot \left(b \cdot c\right)\right)\\ \end{array} \]

Alternative 7
Error	11.2
Cost	1224

\[\begin{array}{l} t_1 := -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\\ \mathbf{if}\;c \leq -4.4 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+136}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	38.0
Cost	1112

\[\begin{array}{l} t_1 := 2 \cdot \left(z \cdot t\right)\\ t_2 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.464755990361542 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -14.02706183642427:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.358512239109047 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.0122104722816374 \cdot 10^{-134}:\\ \;\;\;\;c \cdot \left(i \cdot \left(a \cdot -2\right)\right)\\ \mathbf{elif}\;x \leq 1.7583952725625928 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	37.9
Cost	1112

\[\begin{array}{l} t_1 := 2 \cdot \left(z \cdot t\right)\\ t_2 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.464755990361542 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -14.02706183642427:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.358512239109047 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.906690649013287 \cdot 10^{-146}:\\ \;\;\;\;-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;x \leq 1.7583952725625928 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	37.4
Cost	1108

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot y\right)\\ t_2 := 2 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;t \leq -1.2385999144107699 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+198}:\\ \;\;\;\;i \cdot \left(-2 \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	22.1
Cost	972

\[\begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+155}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(i \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{elif}\;c \leq -2.25 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(c \cdot i\right) \cdot \left(b \cdot c\right)\right)\\ \end{array} \]

Alternative 12
Error	37.3
Cost	848

\[\begin{array}{l} t_1 := 2 \cdot \left(z \cdot t\right)\\ t_2 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.464755990361542 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -14.02706183642427:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.7583952725625928 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	22.0
Cost	840

\[\begin{array}{l} t_1 := -2 \cdot \left(\left(c \cdot i\right) \cdot \left(b \cdot c\right)\right)\\ \mathbf{if}\;c \leq -8 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	22.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+159}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(i \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(c \cdot i\right) \cdot \left(b \cdot c\right)\right)\\ \end{array} \]

Alternative 15
Error	42.8
Cost	320

\[2 \cdot \left(x \cdot y\right) \]

Reproduce

herbie shell --seed 2022291 
(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))))