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

Average Error: 6.3 → 1.7

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t_2 \leq -4.787282543748384 \cdot 10^{+280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2.9417127555116496 \cdot 10^{+258}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(x, y, z \cdot t - i \cdot \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 x y (* z t)) (* c (+ (* c (* b i)) (* a i))))))
        (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -4.787282543748384e+280)
     t_1
     (if (<= t_2 2.9417127555116496e+258)
       (* 2.0 (fma x y (- (* z t) (* i (* c (fma c b a))))))
       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(x, y, (z * t)) - (c * ((c * (b * i)) + (a * i))));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -4.787282543748384e+280) {
		tmp = t_1;
	} else if (t_2 <= 2.9417127555116496e+258) {
		tmp = 2.0 * fma(x, y, ((z * t) - (i * (c * fma(c, b, a)))));
	} 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 * Float64(fma(x, y, Float64(z * t)) - Float64(c * Float64(Float64(c * Float64(b * i)) + Float64(a * i)))))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -4.787282543748384e+280)
		tmp = t_1;
	elseif (t_2 <= 2.9417127555116496e+258)
		tmp = Float64(2.0 * fma(x, y, Float64(Float64(z * t) - Float64(i * Float64(c * fma(c, b, a))))));
	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[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(c * N[(N[(c * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -4.787282543748384e+280], t$95$1, If[LessEqual[t$95$2, 2.9417127555116496e+258], N[(2.0 * N[(x * y + N[(N[(z * t), $MachinePrecision] - N[(i * N[(c * N[(c * b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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	1.8
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.7872825437483836e280 or 2.94171275551164959e258 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 47.2

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

   2. Simplified 47.1

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

   3. Taylor expanded in c around 0 35.0

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

   4. Simplified 15.1

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

   5. Taylor expanded in c around 0 10.5

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

   if -4.7872825437483836e280 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.94171275551164959e258

   1. Initial program 0.4

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

   2. Simplified 0.4

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

   3. Applied egg-rr 0.4

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -4.787282543748384 \cdot 10^{+280}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2.9417127555116496 \cdot 10^{+258}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(x, y, z \cdot t - i \cdot \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022145 
(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))))