Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.7 → 1.3

Time: 7.7s

Precision: binary64

\[[x, y]=\mathsf{sort}([x, y])\]

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \leq -3.0540644172584334 \cdot 10^{-202}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \leq 0:\\ \;\;\;\;\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \leq 8.886014139567274 \cdot 10^{+198}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\sqrt[3]{z}}\\ \end{array}\]

FPCore

(FPCore (x y z t) :precision binary64 (* x (/ (* (/ y z) t) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (/ y z) t) t) (- INFINITY))
   (* (* y x) (/ 1.0 z))
   (if (<= (/ (* (/ y z) t) t) -3.0540644172584334e-202)
     (* (/ y z) x)
     (if (<= (/ (* (/ y z) t) t) 0.0)
       (*
        (* (/ x (* (cbrt z) (cbrt z))) (* (cbrt y) (cbrt y)))
        (/ (cbrt y) (cbrt z)))
       (if (<= (/ (* (/ y z) t) t) 8.886014139567274e+198)
         (* (/ y z) x)
         (/ (/ (* x (/ y (cbrt z))) (cbrt z)) (cbrt z)))))))

C

double code(double x, double y, double z, double t) {
	return x * (((y / z) * t) / t);
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y / z) * t) / t) <= -((double) INFINITY)) {
		tmp = (y * x) * (1.0 / z);
	} else if ((((y / z) * t) / t) <= -3.0540644172584334e-202) {
		tmp = (y / z) * x;
	} else if ((((y / z) * t) / t) <= 0.0) {
		tmp = ((x / (cbrt(z) * cbrt(z))) * (cbrt(y) * cbrt(y))) * (cbrt(y) / cbrt(z));
	} else if ((((y / z) * t) / t) <= 8.886014139567274e+198) {
		tmp = (y / z) * x;
	} else {
		tmp = ((x * (y / cbrt(z))) / cbrt(z)) / cbrt(z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.7
Target	1.5
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (/.f64 y z) t) t) < -inf.0

   1. Initial program 64.0

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 28.4

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied div-inv_binary64_16784 28.5

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{1}{z}\right)}\]

   5. Applied associate-*r*_binary64_16727 2.8

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]

   if -inf.0 < (/.f64 (*.f64 (/.f64 y z) t) t) < -3.0540644172584334e-202 or 0.0 < (/.f64 (*.f64 (/.f64 y z) t) t) < 8.8860141395672744e198

   1. Initial program 0.6

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]

   if -3.0540644172584334e-202 < (/.f64 (*.f64 (/.f64 y z) t) t) < 0.0

   1. Initial program 25.1

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 11.0

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt_binary64_16822 11.3

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity_binary64_16787 11.3

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac_binary64_16793 11.4

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r*_binary64_16727 3.4

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

   8. Simplified 3.4

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_16787 3.4

       \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\color{blue}{1 \cdot \sqrt[3]{z}}}\]

   11. Applied add-cube-cbrt_binary64_16822 3.6

       \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot \sqrt[3]{z}}\]

   12. Applied times-frac_binary64_16793 3.6

       \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)}\]

   13. Applied associate-*r*_binary64_16727 1.4

       \[\leadsto \color{blue}{\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}}\]

   if 8.8860141395672744e198 < (/.f64 (*.f64 (/.f64 y z) t) t)

   1. Initial program 46.2

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 20.1

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt_binary64_16822 20.9

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity_binary64_16787 20.9

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac_binary64_16793 20.9

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r*_binary64_16727 7.4

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

   8. Simplified 7.4

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt_binary64_16822 7.6

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\]

   11. Applied times-frac_binary64_16793 7.6

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)} \cdot \frac{y}{\sqrt[3]{z}}\]

   12. Applied associate-*l*_binary64_16728 6.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   13. Simplified 6.6

       \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \color{blue}{\left(\frac{y}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)}\]
   14. Using strategy rm

   15. Applied associate-*l/_binary64_16730 7.0

       \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{y}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)}{\sqrt[3]{z}}}\]

   16. Simplified 7.1

       \[\leadsto \frac{\color{blue}{\frac{x \cdot \frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}}}}{\sqrt[3]{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \leq -3.0540644172584334 \cdot 10^{-202}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \leq 0:\\ \;\;\;\;\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \leq 8.886014139567274 \cdot 10^{+198}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))