Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 83.9% → 96.7%

Time: 8.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y - z\right)}{t - z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y - z\right)}{t - z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y - z}{t - z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[\frac{x \cdot \left(y - z\right)}{t - z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.2

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

4. Applied rewrites 96.2%

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

5. Final simplification 96.2%

   \[\leadsto x \cdot \frac{y - z}{t - z} \]
6. Add Preprocessing

Alternative 2: 58.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -20:\\ \;\;\;\;\frac{y}{t - z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{t} \cdot \left(y - z\right)\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{x \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -20.0)
     (* (/ y (- t z)) x)
     (if (<= t_1 0.0)
       (* (/ x t) (- y z))
       (if (<= t_1 1.5e+89) (/ (* x z) (- z t)) (fma (- y) (/ x z) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -20.0) {
		tmp = (y / (t - z)) * x;
	} else if (t_1 <= 0.0) {
		tmp = (x / t) * (y - z);
	} else if (t_1 <= 1.5e+89) {
		tmp = (x * z) / (z - t);
	} else {
		tmp = fma(-y, (x / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -20.0)
		tmp = Float64(Float64(y / Float64(t - z)) * x);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x / t) * Float64(y - z));
	elseif (t_1 <= 1.5e+89)
		tmp = Float64(Float64(x * z) / Float64(z - t));
	else
		tmp = fma(Float64(-y), Float64(x / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20.0], N[(N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+89], N[(N[(x * z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -20

   1. Initial program 71.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{t - z}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if -20 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -0.0

   1. Initial program 95.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{t}} \cdot \left(y - z\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 57.0

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

   7. Applied rewrites 57.0%

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

   if -0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.50000000000000006e89

   1. Initial program 99.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{y - z}{t - z} \cdot x} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t - z}{y - z}}} \cdot x \]

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{-1}{\color{blue}{0 - \left(t - z\right)}} \cdot \left(y - z\right)\right) \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{-1}{0 - \color{blue}{\left(t - z\right)}} \cdot \left(y - z\right)\right) \cdot x \]

      10. sub-neg N/A

          \[\leadsto \left(\frac{-1}{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(z\right)\right)\right)}} \cdot \left(y - z\right)\right) \cdot x \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{-1}{0 - \left(t + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)} \cdot \left(y - z\right)\right) \cdot x \]

      12. +-commutative N/A

          \[\leadsto \left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + t\right)}} \cdot \left(y - z\right)\right) \cdot x \]

      13. associate--r+ N/A

          \[\leadsto \left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - t}} \cdot \left(y - z\right)\right) \cdot x \]

      14. neg-sub0 N/A

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

      15. lift-neg.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 96.5

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

   6. Applied rewrites 96.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot z}{z - t}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 58.6

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

   9. Applied rewrites 58.6%

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

   if 1.50000000000000006e89 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 62.3%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y - z\right)}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x \cdot \left(y - z\right)}{z}} \]

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{z}{z}\right)\right)\right)} \]

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto 0 - \left(\frac{x \cdot y}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto 0 - \color{blue}{\left(\frac{x \cdot y}{z} - x\right)} \]

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\left(0 - \frac{x \cdot y}{z}\right) + x} \]

      14. neg-sub0 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. distribute-lft-neg-in N/A

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 63.6

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

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, \frac{x}{z}, x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Developer Target 1: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ (- t z) (- y z))))

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