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

Average Error: 10.4 → 0.1

Time: 11.8s

Precision: binary64

Cost: 904

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;y + \frac{x + y \cdot \left(-x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e+61)
   (* (- 1.0 (/ x z)) y)
   (if (<= y 1.6e+16) (+ y (/ (+ x (* y (- x))) z)) (- y (* y (/ x z))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+61) {
		tmp = (1.0 - (x / z)) * y;
	} else if (y <= 1.6e+16) {
		tmp = y + ((x + (y * -x)) / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.6d+61)) then
        tmp = (1.0d0 - (x / z)) * y
    else if (y <= 1.6d+16) then
        tmp = y + ((x + (y * -x)) / z)
    else
        tmp = y - (y * (x / z))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+61) {
		tmp = (1.0 - (x / z)) * y;
	} else if (y <= 1.6e+16) {
		tmp = y + ((x + (y * -x)) / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

↓

def code(x, y, z):
	tmp = 0
	if y <= -1.6e+61:
		tmp = (1.0 - (x / z)) * y
	elif y <= 1.6e+16:
		tmp = y + ((x + (y * -x)) / z)
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e+61)
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	elseif (y <= 1.6e+16)
		tmp = Float64(y + Float64(Float64(x + Float64(y * Float64(-x))) / z));
	else
		tmp = Float64(y - Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e+61)
		tmp = (1.0 - (x / z)) * y;
	elseif (y <= 1.6e+16)
		tmp = y + ((x + (y * -x)) / z);
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, -1.6e+61], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.6e+16], N[(y + N[(N[(x + N[(y * (-x)), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.4
Target	0.0
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -1.5999999999999999e61

   1. Initial program 28.8

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

   2. Taylor expanded in x around inf 9.2

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

   3. Simplified 9.2

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

      [Start] 9.2	\[ y + \frac{\left(1 + -1 \cdot y\right) \cdot x}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 9.2	\[ y + \frac{\color{blue}{x \cdot \left(1 + -1 \cdot y\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 9.2	\[ y + \frac{\color{blue}{1 \cdot x + x \cdot \left(-1 \cdot y\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 9.2	\[ y + \frac{\color{blue}{x \cdot 1} + x \cdot \left(-1 \cdot y\right)}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 9.2	\[ y + \frac{\color{blue}{x} + x \cdot \left(-1 \cdot y\right)}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 9.2	\[ y + \frac{x + x \cdot \color{blue}{\left(y \cdot -1\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [<=] 9.2	\[ y + \frac{x + \color{blue}{y \cdot \left(x \cdot -1\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 9.2	\[ y + \frac{x + y \cdot \color{blue}{\left(-x\right)}}{z} \]

   4. Taylor expanded in y around inf 0.1

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

   5. Simplified 0.1

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

      [Start] 0.1	\[ y \cdot \left(1 + -1 \cdot \frac{x}{z}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 0.1	\[ \color{blue}{1 \cdot y + y \cdot \left(-1 \cdot \frac{x}{z}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.1	\[ \color{blue}{y \cdot 1} + y \cdot \left(-1 \cdot \frac{x}{z}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.1	\[ \color{blue}{y} + y \cdot \left(-1 \cdot \frac{x}{z}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ y + y \cdot \color{blue}{\left(\frac{x}{z} \cdot -1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.1	\[ y + y \cdot \color{blue}{\left(-\frac{x}{z}\right)} \]

   6. Applied egg-rr 0.1

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

   if -1.5999999999999999e61 < y < 1.6e16

   1. Initial program 0.5

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

   2. Taylor expanded in x around inf 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ y + \frac{\left(1 + -1 \cdot y\right) \cdot x}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ y + \frac{\color{blue}{x \cdot \left(1 + -1 \cdot y\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 0.1	\[ y + \frac{\color{blue}{1 \cdot x + x \cdot \left(-1 \cdot y\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.1	\[ y + \frac{\color{blue}{x \cdot 1} + x \cdot \left(-1 \cdot y\right)}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.1	\[ y + \frac{\color{blue}{x} + x \cdot \left(-1 \cdot y\right)}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ y + \frac{x + x \cdot \color{blue}{\left(y \cdot -1\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [<=] 0.1	\[ y + \frac{x + \color{blue}{y \cdot \left(x \cdot -1\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 0.1	\[ y + \frac{x + y \cdot \color{blue}{\left(-x\right)}}{z} \]

   if 1.6e16 < y

   1. Initial program 24.5

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

   2. Taylor expanded in x around inf 8.0

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

   3. Simplified 8.0

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

      [Start] 8.0	\[ y + \frac{\left(1 + -1 \cdot y\right) \cdot x}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 8.0	\[ y + \frac{\color{blue}{x \cdot \left(1 + -1 \cdot y\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 8.0	\[ y + \frac{\color{blue}{1 \cdot x + x \cdot \left(-1 \cdot y\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 8.0	\[ y + \frac{\color{blue}{x \cdot 1} + x \cdot \left(-1 \cdot y\right)}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 8.0	\[ y + \frac{\color{blue}{x} + x \cdot \left(-1 \cdot y\right)}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 8.0	\[ y + \frac{x + x \cdot \color{blue}{\left(y \cdot -1\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [<=] 8.0	\[ y + \frac{x + \color{blue}{y \cdot \left(x \cdot -1\right)}}{z} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 8.0	\[ y + \frac{x + y \cdot \color{blue}{\left(-x\right)}}{z} \]

   4. Taylor expanded in y around inf 0.1

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

   5. Simplified 0.1

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

      [Start] 0.1	\[ y \cdot \left(1 + -1 \cdot \frac{x}{z}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 0.1	\[ \color{blue}{1 \cdot y + y \cdot \left(-1 \cdot \frac{x}{z}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.1	\[ \color{blue}{y \cdot 1} + y \cdot \left(-1 \cdot \frac{x}{z}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.1	\[ \color{blue}{y} + y \cdot \left(-1 \cdot \frac{x}{z}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ y + y \cdot \color{blue}{\left(\frac{x}{z} \cdot -1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.1	\[ y + y \cdot \color{blue}{\left(-\frac{x}{z}\right)} \]

   6. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;y + \frac{x + y \cdot \left(-x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.15 \cdot 10^{+14}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	712

\[\begin{array}{l} t_0 := \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+37}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	0.0
Cost	704

\[\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right) \]

Alternative 5
Error	20.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	9.0
Cost	320

\[\frac{x}{z} + y \]

Alternative 7
Error	31.5
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023090 
(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))