Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Average Error: 16.25% → 0.32%

Time: 7.2s

Precision: binary64

Cost: 8136

Math

\[\frac{x + y \cdot \left(z - x\right)}{z} \]

↓

\[\begin{array}{l} t_0 := \frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x (* y (- z x))) z)))
   (if (<= t_0 (- INFINITY))
     (* y (/ (- z x) z))
     (if (<= t_0 2e+301) (/ (fma y (- z x) x) z) (- y (/ y (/ z x)))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x + (y * (z - x))) / z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = y * ((z - x) / z);
	} else if (t_0 <= 2e+301) {
		tmp = fma(y, (z - x), x) / z;
	} else {
		tmp = y - (y / (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * Float64(z - x))) / z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(y * Float64(Float64(z - x) / z));
	elseif (t_0 <= 2e+301)
		tmp = Float64(fma(y, Float64(z - x), x) / z);
	else
		tmp = Float64(y - Float64(y / Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+301], N[(N[(y * N[(z - x), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	16.25%
Target	0.07%
Herbie	0.32%

\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Simplified 100

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}} \]
      Proof

      [Start] 100	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
      +-commutative [=>] 100	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
      fma-def [=>] 100	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

   3. Taylor expanded in y around inf 100

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

   4. Simplified 0.14

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

      [Start] 100	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      *-commutative [=>] 100	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      associate-/l* [=>] 0.26	\[ \color{blue}{\frac{z - x}{\frac{z}{y}}} \]
      associate-/r/ [=>] 0.14	\[ \color{blue}{\frac{z - x}{z} \cdot y} \]
      remove-double-neg [<=] 0.14	\[ \frac{\color{blue}{-\left(-\left(z - x\right)\right)}}{z} \cdot y \]
      neg-mul-1 [=>] 0.14	\[ \frac{-\color{blue}{-1 \cdot \left(z - x\right)}}{z} \cdot y \]
      *-commutative [=>] 0.14	\[ \frac{-\color{blue}{\left(z - x\right) \cdot -1}}{z} \cdot y \]
      distribute-rgt-neg-in [=>] 0.14	\[ \frac{\color{blue}{\left(z - x\right) \cdot \left(--1\right)}}{z} \cdot y \]
      metadata-eval [=>] 0.14	\[ \frac{\left(z - x\right) \cdot \color{blue}{1}}{z} \cdot y \]
      associate-*r/ [<=] 0.36	\[ \color{blue}{\left(\left(z - x\right) \cdot \frac{1}{z}\right)} \cdot y \]
      unpow-1 [<=] 0.36	\[ \left(\left(z - x\right) \cdot \color{blue}{{z}^{-1}}\right) \cdot y \]
      *-commutative [<=] 0.36	\[ \color{blue}{y \cdot \left(\left(z - x\right) \cdot {z}^{-1}\right)} \]
      unpow-1 [=>] 0.36	\[ y \cdot \left(\left(z - x\right) \cdot \color{blue}{\frac{1}{z}}\right) \]
      associate-*r/ [=>] 0.14	\[ y \cdot \color{blue}{\frac{\left(z - x\right) \cdot 1}{z}} \]
      metadata-eval [<=] 0.14	\[ y \cdot \frac{\left(z - x\right) \cdot \color{blue}{\left(--1\right)}}{z} \]
      distribute-rgt-neg-in [<=] 0.14	\[ y \cdot \frac{\color{blue}{-\left(z - x\right) \cdot -1}}{z} \]
      *-commutative [<=] 0.14	\[ y \cdot \frac{-\color{blue}{-1 \cdot \left(z - x\right)}}{z} \]
      neg-mul-1 [<=] 0.14	\[ y \cdot \frac{-\color{blue}{\left(-\left(z - x\right)\right)}}{z} \]
      remove-double-neg [=>] 0.14	\[ y \cdot \frac{\color{blue}{z - x}}{z} \]

   if -inf.0 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < 2.00000000000000011e301

   1. Initial program 0.14

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Simplified 0.14

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}} \]
      Proof

      [Start] 0.14	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
      +-commutative [=>] 0.14	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
      fma-def [=>] 0.14	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z)

   1. Initial program 95.4

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Simplified 95.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}} \]
      Proof

      [Start] 95.4	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
      +-commutative [=>] 95.4	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
      fma-def [=>] 95.4	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

   3. Taylor expanded in y around inf 97.48

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

   4. Simplified 2.18

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

      [Start] 97.48	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      *-commutative [=>] 97.48	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      associate-/l* [=>] 3.7	\[ \color{blue}{\frac{z - x}{\frac{z}{y}}} \]
      associate-/r/ [=>] 2.18	\[ \color{blue}{\frac{z - x}{z} \cdot y} \]
      remove-double-neg [<=] 2.18	\[ \frac{\color{blue}{-\left(-\left(z - x\right)\right)}}{z} \cdot y \]
      neg-mul-1 [=>] 2.18	\[ \frac{-\color{blue}{-1 \cdot \left(z - x\right)}}{z} \cdot y \]
      *-commutative [=>] 2.18	\[ \frac{-\color{blue}{\left(z - x\right) \cdot -1}}{z} \cdot y \]
      distribute-rgt-neg-in [=>] 2.18	\[ \frac{\color{blue}{\left(z - x\right) \cdot \left(--1\right)}}{z} \cdot y \]
      metadata-eval [=>] 2.18	\[ \frac{\left(z - x\right) \cdot \color{blue}{1}}{z} \cdot y \]
      associate-*r/ [<=] 2.42	\[ \color{blue}{\left(\left(z - x\right) \cdot \frac{1}{z}\right)} \cdot y \]
      unpow-1 [<=] 2.42	\[ \left(\left(z - x\right) \cdot \color{blue}{{z}^{-1}}\right) \cdot y \]
      *-commutative [<=] 2.42	\[ \color{blue}{y \cdot \left(\left(z - x\right) \cdot {z}^{-1}\right)} \]
      unpow-1 [=>] 2.42	\[ y \cdot \left(\left(z - x\right) \cdot \color{blue}{\frac{1}{z}}\right) \]
      associate-*r/ [=>] 2.18	\[ y \cdot \color{blue}{\frac{\left(z - x\right) \cdot 1}{z}} \]
      metadata-eval [<=] 2.18	\[ y \cdot \frac{\left(z - x\right) \cdot \color{blue}{\left(--1\right)}}{z} \]
      distribute-rgt-neg-in [<=] 2.18	\[ y \cdot \frac{\color{blue}{-\left(z - x\right) \cdot -1}}{z} \]
      *-commutative [<=] 2.18	\[ y \cdot \frac{-\color{blue}{-1 \cdot \left(z - x\right)}}{z} \]
      neg-mul-1 [<=] 2.18	\[ y \cdot \frac{-\color{blue}{\left(-\left(z - x\right)\right)}}{z} \]
      remove-double-neg [=>] 2.18	\[ y \cdot \frac{\color{blue}{z - x}}{z} \]

   5. Taylor expanded in y around 0 97.48

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

   6. Simplified 2.17

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

      [Start] 97.48	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      *-commutative [=>] 97.48	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      associate-/l* [=>] 3.7	\[ \color{blue}{\frac{z - x}{\frac{z}{y}}} \]
      div-sub [=>] 3.69	\[ \color{blue}{\frac{z}{\frac{z}{y}} - \frac{x}{\frac{z}{y}}} \]
      associate-/r/ [=>] 3.55	\[ \color{blue}{\frac{z}{z} \cdot y} - \frac{x}{\frac{z}{y}} \]
      remove-double-neg [<=] 3.55	\[ \frac{\color{blue}{-\left(-z\right)}}{z} \cdot y - \frac{x}{\frac{z}{y}} \]
      neg-mul-1 [=>] 3.55	\[ \frac{\color{blue}{-1 \cdot \left(-z\right)}}{z} \cdot y - \frac{x}{\frac{z}{y}} \]
      distribute-rgt-neg-in [<=] 3.55	\[ \frac{\color{blue}{--1 \cdot z}}{z} \cdot y - \frac{x}{\frac{z}{y}} \]
      distribute-lft-neg-in [=>] 3.55	\[ \frac{\color{blue}{\left(--1\right) \cdot z}}{z} \cdot y - \frac{x}{\frac{z}{y}} \]
      metadata-eval [=>] 3.55	\[ \frac{\color{blue}{1} \cdot z}{z} \cdot y - \frac{x}{\frac{z}{y}} \]
      associate-*l/ [<=] 3.78	\[ \color{blue}{\left(\frac{1}{z} \cdot z\right)} \cdot y - \frac{x}{\frac{z}{y}} \]
      lft-mult-inverse [=>] 3.55	\[ \color{blue}{1} \cdot y - \frac{x}{\frac{z}{y}} \]
      *-lft-identity [=>] 3.55	\[ \color{blue}{y} - \frac{x}{\frac{z}{y}} \]
      associate-/l* [<=] 32.68	\[ y - \color{blue}{\frac{x \cdot y}{z}} \]
      *-commutative [<=] 32.68	\[ y - \frac{\color{blue}{y \cdot x}}{z} \]
      associate-/l* [=>] 2.17	\[ y - \color{blue}{\frac{y}{\frac{z}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.32

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.33%
Cost	1864

\[\begin{array}{l} t_0 := \frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 2
Error	20.43%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-25} \lor \neg \left(y \leq -7.6 \cdot 10^{-174} \lor \neg \left(y \leq -8.2 \cdot 10^{-228}\right) \land y \leq 8.2 \cdot 10^{-16}\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Error	20.61%
Cost	977

\[\begin{array}{l} t_0 := y \cdot \frac{z - x}{z}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-174}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-228} \lor \neg \left(y \leq 22000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 4
Error	39.88%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-91}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+46}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5
Error	49.22%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))