Average Error: 1.9 → 1.4
Time: 8.1s
Precision: binary64
Cost: 7108
\[x + \left(y - x\right) \cdot \frac{z}{t} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+243}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]
(FPCore (x y z t) :precision binary64 (+ x (* (- y x) (/ z t))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -2e+243) (* z (/ (- y x) t)) (fma (- y x) (/ z t) x)))
double code(double x, double y, double z, double t) {
	return x + ((y - x) * (z / t));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -2e+243) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = fma((y - x), (z / t), x);
	}
	return tmp;
}

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - x) * Float64(z / t)))
end
↓
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= -2e+243)
		tmp = Float64(z * Float64(Float64(y - x) / t));
	else
		tmp = fma(Float64(y - x), Float64(z / t), x);
	end
	return tmp
end

code[x_, y_, z_, t_] := N[(x + N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], -2e+243], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+243}:\\
\;\;\;\;z \cdot \frac{y - x}{t}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\


\end{array}

Error

Target

Original1.9
Target2.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 z t) < -2.0000000000000001e243

    1. Initial program 27.1

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

    2. Simplified27.1

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

      [Start]27.1

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

      +-commutative [=>]27.1

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

      fma-def [=>]27.1

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

    3. Taylor expanded in z around inf 0.9

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

    4. Simplified0.9

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

      [Start]0.9

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

      *-commutative [=>]0.9

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

      div-sub [<=]0.9

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

    if -2.0000000000000001e243 < (/.f64 z t)

    1. Initial program 1.4

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

    2. Simplified1.4

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

      [Start]1.4

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

      +-commutative [=>]1.4

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

      fma-def [=>]1.4

      \[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+243}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error21.9
Cost1684
\[\begin{array}{l} t_1 := \frac{-x}{\frac{t}{z}}\\ t_2 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq -4 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error21.7
Cost1164
\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \end{array} \]
Alternative 3
Error14.3
Cost969
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-43} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 4
Error4.6
Cost969
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+24} \lor \neg \left(\frac{z}{t} \leq 10^{-5}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \]
Alternative 5
Error3.1
Cost969
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -10000000 \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{y - x}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \]
Alternative 6
Error21.6
Cost841
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-43} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 7
Error21.5
Cost841
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-43} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 8
Error1.4
Cost836
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+243}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \end{array} \]
Alternative 9
Error31.1
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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