Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Average Accuracy: 82.6% → 97.5%

Time: 16.0s

Precision: binary64

Cost: 7620

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;x - \frac{t \cdot \left(z - y\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{z - y}{a - z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) t) (- a z))))
   (if (<= t_1 0.0)
     (fma (- y z) (/ t (- a z)) x)
     (if (<= t_1 2e+284)
       (- x (/ (* t (- z y)) (- a z)))
       (- x (* t (/ (- z y) (- a z))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (t_1 <= 2e+284) {
		tmp = x - ((t * (z - y)) / (a - z));
	} else {
		tmp = x - (t * ((z - y) / (a - z)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), x);
	elseif (t_1 <= 2e+284)
		tmp = Float64(x - Float64(Float64(t * Float64(z - y)) / Float64(a - z)));
	else
		tmp = Float64(x - Float64(t * Float64(Float64(z - y) / Float64(a - z))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+284], N[(x - N[(N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t * N[(N[(z - y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	82.6%
Target	99.2%
Herbie	97.5%

\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 0.0

   1. Initial program 84.5%

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

   2. Simplified 95.8%

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

      [Start] 84.5	\[ x + \frac{\left(y - z\right) \cdot t}{a - z} \]
      +-commutative [=>] 84.5	\[ \color{blue}{\frac{\left(y - z\right) \cdot t}{a - z} + x} \]
      associate-*r/ [<=] 95.8	\[ \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} + x \]
      fma-def [=>] 95.8	\[ \color{blue}{\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)} \]

   if 0.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 2.00000000000000016e284

   1. Initial program 99.8%

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

   if 2.00000000000000016e284 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 5.4%

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

   2. Simplified 98.7%

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

      [Start] 5.4	\[ x + \frac{\left(y - z\right) \cdot t}{a - z} \]
      associate-*l/ [<=] 98.7	\[ x + \color{blue}{\frac{y - z}{a - z} \cdot t} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 2 \cdot 10^{+284}:\\ \;\;\;\;x - \frac{t \cdot \left(z - y\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{z - y}{a - z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1993

\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+284}\right):\\ \;\;\;\;x - t \cdot \frac{z - y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot \left(z - y\right)}{a - z}\\ \end{array} \]

Alternative 2
Accuracy	65.5%
Cost	1765

\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ t_2 := \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.86 \cdot 10^{-39}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-239}:\\ \;\;\;\;x - \frac{y}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.56 \cdot 10^{+72} \lor \neg \left(t \leq 1.2 \cdot 10^{+222}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 3
Accuracy	63.9%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-133}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-191}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-232}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 4
Accuracy	97.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-164} \lor \neg \left(t \leq 5 \cdot 10^{-187}\right):\\ \;\;\;\;x - t \cdot \frac{z - y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \end{array} \]

Alternative 5
Accuracy	65.8%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+34}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-236}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-245}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 6
Accuracy	82.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+205} \lor \neg \left(z \leq 5.5 \cdot 10^{+118}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \end{array} \]

Alternative 7
Accuracy	84.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+33} \lor \neg \left(z \leq 7 \cdot 10^{+50}\right):\\ \;\;\;\;x + \left(z - y\right) \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \end{array} \]

Alternative 8
Accuracy	86.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-41} \lor \neg \left(y \leq 4.4 \cdot 10^{+123}\right):\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{z}{a - z}\\ \end{array} \]

Alternative 9
Accuracy	86.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+33} \lor \neg \left(z \leq 3.8 \cdot 10^{+50}\right):\\ \;\;\;\;x + t \cdot \frac{z - y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \end{array} \]

Alternative 10
Accuracy	67.8%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+32}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-293}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-249}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 11
Accuracy	77.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.245:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+44}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 12
Accuracy	77.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 13
Accuracy	57.0%
Cost	592

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-101}:\\ \;\;\;\;t\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-201}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-135}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Accuracy	68.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]

Alternative 15
Accuracy	20.1%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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