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

Average Error: 11.6 → 1.5

Time: 11.8s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ t_2 := \frac{z - y}{\frac{z - t}{x}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+271}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))) (t_2 (/ (- z y) (/ (- z t) x))))
   (if (<= t_1 -1e+271) t_2 (if (<= t_1 4e+219) t_1 t_2))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double t_2 = (z - y) / ((z - t) / x);
	double tmp;
	if (t_1 <= -1e+271) {
		tmp = t_2;
	} else if (t_1 <= 4e+219) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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

↓

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * (y - z)) / (t - z)
    t_2 = (z - y) / ((z - t) / x)
    if (t_1 <= (-1d+271)) then
        tmp = t_2
    else if (t_1 <= 4d+219) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double t_2 = (z - y) / ((z - t) / x);
	double tmp;
	if (t_1 <= -1e+271) {
		tmp = t_2;
	} else if (t_1 <= 4e+219) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (x * (y - z)) / (t - z)
	t_2 = (z - y) / ((z - t) / x)
	tmp = 0
	if t_1 <= -1e+271:
		tmp = t_2
	elif t_1 <= 4e+219:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	t_2 = Float64(Float64(z - y) / Float64(Float64(z - t) / x))
	tmp = 0.0
	if (t_1 <= -1e+271)
		tmp = t_2;
	elseif (t_1 <= 4e+219)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (x * (y - z)) / (t - z);
	t_2 = (z - y) / ((z - t) / x);
	tmp = 0.0;
	if (t_1 <= -1e+271)
		tmp = t_2;
	elseif (t_1 <= 4e+219)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - y), $MachinePrecision] / N[(N[(z - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+271], t$95$2, If[LessEqual[t$95$1, 4e+219], t$95$1, t$95$2]]]]

Target

Original	11.6
Target	2.2
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -9.99999999999999953e270 or 3.99999999999999986e219 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 54.2

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

   2. Simplified 2.7

      \[\leadsto \color{blue}{\left(z - y\right) \cdot \frac{x}{z - t}} \]
      Proof
      (*.f64 (-.f64 z y) (/.f64 x (-.f64 z t))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 x (-.f64 z t)) (-.f64 z y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 x (/.f64 (-.f64 z t) (-.f64 z y)))): 24 points increase in error, 71 points decrease in error
      (/.f64 x (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 z t) (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 z t) (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 z t)) (*.f64 -1 (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z t))) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z t))) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) t)) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) t) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 z))) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 t z)) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= sub-neg_binary64 (-.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))): 73 points increase in error, 20 points decrease in error

   3. Applied egg-rr 2.4

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

   if -9.99999999999999953e270 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 3.99999999999999986e219

   1. Initial program 1.3

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{+271}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 4 \cdot 10^{+219}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.9
Cost	1372

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ t_2 := x - x \cdot \frac{y}{z}\\ t_3 := \frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{if}\;z \leq -1.9932437566209087 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.897223780106938 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.427013916189279 \cdot 10^{-14}:\\ \;\;\;\;\frac{z}{\frac{z - t}{x}}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.8
Cost	1372

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ t_2 := x - x \cdot \frac{y}{z}\\ t_3 := \frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{if}\;z \leq -1.9932437566209087 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.897223780106938 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.427013916189279 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	16.9
Cost	1108

\[\begin{array}{l} t_1 := x - x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.9932437566209087 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.897223780106938 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq -5.427013916189279 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.7
Cost	1104

\[\begin{array}{l} t_1 := \frac{z - y}{\frac{z - t}{x}}\\ t_2 := x - x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -2.189679110726013 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-136}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 5.35606196721939 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	7.8
Cost	840

\[\begin{array}{l} t_1 := x - x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -8.867411073041705 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.35606196721939 \cdot 10^{+229}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	21.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9932437566209087 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 43827659581.32647:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	17.5
Cost	712

\[\begin{array}{l} t_1 := x - x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.9932437566209087 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	2.3
Cost	704

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

Alternative 9
Error	37.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3.0007989324458855 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;\frac{z}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	25.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.867411073041705 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;\frac{y}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	24.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.867411073041705 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	24.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.867411073041705 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 801815.5011172196:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	39.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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