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

Average Error: 6.0 → 1.2

Time: 10.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y + z \cdot t\\ t_2 := a + b \cdot c\\ t_3 := \left(c \cdot t_2\right) \cdot i\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;2 \cdot \left(t_1 - t_2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;t_3 \leq 5.29042202284554 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \left(t_1 - t_3\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt{c \cdot \left(t_2 \cdot i\right)}\\ 2 \cdot \left(t_1 - t_4 \cdot t_4\right) \end{array}\\ \end{array} \]

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 (+ (* x y) (* z t))) (t_2 (+ a (* b c))) (t_3 (* (* c t_2) i)))
   (if (<= t_3 (- INFINITY))
     (* 2.0 (- t_1 (* t_2 (* c i))))
     (if (<= t_3 5.29042202284554e+307)
       (* 2.0 (- t_1 t_3))
       (let* ((t_4 (sqrt (* c (* t_2 i))))) (* 2.0 (- t_1 (* t_4 t_4))))))))

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 = (x * y) + (z * t);
	double t_2 = a + (b * c);
	double t_3 = (c * t_2) * i;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = 2.0 * (t_1 - (t_2 * (c * i)));
	} else if (t_3 <= 5.29042202284554e+307) {
		tmp = 2.0 * (t_1 - t_3);
	} else {
		double t_4 = sqrt(c * (t_2 * i));
		tmp = 2.0 * (t_1 - (t_4 * t_4));
	}
	return tmp;
}

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	6.0
Target	1.8
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   3. Applied associate-*l*_binary64 9.9

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.29042202284553989e307

   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. Using strategy rm

   3. Applied add-cube-cbrt_binary64 0.6

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(\sqrt[3]{\left(a + b \cdot c\right) \cdot c} \cdot \sqrt[3]{\left(a + b \cdot c\right) \cdot c}\right) \cdot \sqrt[3]{\left(a + b \cdot c\right) \cdot c}\right)} \cdot i\right) \]

   4. Simplified 0.6

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{\left(\sqrt[3]{c \cdot \left(a + b \cdot c\right)} \cdot \sqrt[3]{c \cdot \left(a + b \cdot c\right)}\right)} \cdot \sqrt[3]{\left(a + b \cdot c\right) \cdot c}\right) \cdot i\right) \]

   5. Simplified 0.6

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(\sqrt[3]{c \cdot \left(a + b \cdot c\right)} \cdot \sqrt[3]{c \cdot \left(a + b \cdot c\right)}\right) \cdot \color{blue}{\sqrt[3]{c \cdot \left(a + b \cdot c\right)}}\right) \cdot i\right) \]
   6. Using strategy rm

   7. Applied *-un-lft-identity_binary64 0.6

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(\sqrt[3]{c \cdot \left(a + b \cdot c\right)} \cdot \sqrt[3]{c \cdot \left(a + b \cdot c\right)}\right) \cdot \sqrt[3]{c \cdot \left(a + b \cdot c\right)}\right) \cdot \color{blue}{\left(1 \cdot i\right)}\right) \]

   8. Applied associate-*r*_binary64 0.6

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(\left(\sqrt[3]{c \cdot \left(a + b \cdot c\right)} \cdot \sqrt[3]{c \cdot \left(a + b \cdot c\right)}\right) \cdot \sqrt[3]{c \cdot \left(a + b \cdot c\right)}\right) \cdot 1\right) \cdot i}\right) \]

   9. Simplified 0.3

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

   if 5.29042202284553989e307 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 63.8

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

   3. Applied add-sqr-sqrt_binary64 63.8

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

   4. Simplified 63.8

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

   5. Simplified 10.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -\infty:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5.29042202284554 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \sqrt{c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)} \cdot \sqrt{c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021190 
(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))))