Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 24.2 → 9.0

Time: 26.8s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -7.063120766029758 \cdot 10^{+157}:\\ \;\;\;\;y - \left(y - x\right) \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t \leq 7.239090729937892 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(\frac{a}{t} + 1\right) \cdot \left(\frac{y - x}{t} \cdot \left(a - z\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.063120766029758e+157)
   (- y (* (- y x) (/ (- z a) t)))
   (if (<= t 7.239090729937892e+72)
     (fma (- y x) (/ (- z t) (- a t)) x)
     (+ y (* (+ (/ a t) 1.0) (* (/ (- y x) t) (- a z)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.063120766029758e+157) {
		tmp = y - ((y - x) * ((z - a) / t));
	} else if (t <= 7.239090729937892e+72) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = y + (((a / t) + 1.0) * (((y - x) / t) * (a - z)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.063120766029758e+157)
		tmp = Float64(y - Float64(Float64(y - x) * Float64(Float64(z - a) / t)));
	elseif (t <= 7.239090729937892e+72)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = Float64(y + Float64(Float64(Float64(a / t) + 1.0) * Float64(Float64(Float64(y - x) / t) * Float64(a - z))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.063120766029758e+157], N[(y - N[(N[(y - x), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.239090729937892e+72], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(N[(N[(a / t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	24.2
Target	9.1
Herbie	9.0

\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if t < -7.0631207660297579e157

   1. Initial program 47.1

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

   2. Simplified 22.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
      Proof
      (fma.f64 (-.f64 y x) (/.f64 (-.f64 z t) (-.f64 a t)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) (/.f64 (-.f64 z t) (-.f64 a t))) x)): 2 points increase in error, 4 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) x): 114 points increase in error, 10 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around inf 25.2

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

   4. Simplified 9.7

      \[\leadsto \color{blue}{y + \frac{a - z}{t} \cdot \left(y - x\right)} \]
      Proof
      (+.f64 y (*.f64 (/.f64 (-.f64 a z) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 a (neg.f64 z))) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 (/.f64 (+.f64 a (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) a)) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 (/.f64 (+.f64 (*.f64 -1 z) (Rewrite<= *-lft-identity_binary64 (*.f64 1 a))) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 (/.f64 (+.f64 (*.f64 -1 z) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) a)) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 -1 z) (*.f64 -1 a))) t) (-.f64 y x))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (*.f64 -1 z) (*.f64 -1 a)) (/.f64 t (-.f64 y x))))): 24 points increase in error, 13 points decrease in error
      (+.f64 y (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (*.f64 -1 z) (*.f64 -1 a)) (-.f64 y x)) t))): 38 points increase in error, 12 points decrease in error
      (+.f64 y (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 y x) (-.f64 (*.f64 -1 z) (*.f64 -1 a)))) t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (-.f64 y x) (-.f64 (*.f64 -1 z) (*.f64 -1 a))) t) y)): 0 points increase in error, 0 points decrease in error

   if -7.0631207660297579e157 < t < 7.2390907299378918e72

   1. Initial program 13.8

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

   2. Simplified 7.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
      Proof
      (fma.f64 (-.f64 y x) (/.f64 (-.f64 z t) (-.f64 a t)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) (/.f64 (-.f64 z t) (-.f64 a t))) x)): 2 points increase in error, 4 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) x): 114 points increase in error, 10 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))): 0 points increase in error, 0 points decrease in error

   if 7.2390907299378918e72 < t

   1. Initial program 41.7

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

   2. Taylor expanded in t around inf 29.8

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \left(\frac{\left(-1 \cdot \left(\left(y - x\right) \cdot z\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right) \cdot a}{{t}^{2}} + y\right)\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

   3. Simplified 14.9

      \[\leadsto \color{blue}{y - \left(\frac{a}{t} + 1\right) \cdot \left(\frac{y - x}{t} \cdot \left(z - a\right)\right)} \]
      Proof
      (-.f64 y (*.f64 (+.f64 (/.f64 a t) 1) (*.f64 (/.f64 (-.f64 y x) t) (-.f64 z a)))): 0 points increase in error, 0 points decrease in error
      (-.f64 y (*.f64 (+.f64 (/.f64 a t) 1) (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 y x) (/.f64 t (-.f64 z a)))))): 14 points increase in error, 17 points decrease in error
      (-.f64 y (*.f64 (+.f64 (/.f64 a t) 1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 y x) (-.f64 z a)) t)))): 39 points increase in error, 11 points decrease in error
      (-.f64 y (*.f64 (+.f64 (/.f64 a t) 1) (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x)))) t))): 0 points increase in error, 2 points decrease in error
      (-.f64 y (*.f64 (+.f64 (/.f64 a t) 1) (/.f64 (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 y x) z)) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (-.f64 y (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t) (*.f64 (/.f64 a t) (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))))): 0 points increase in error, 0 points decrease in error
      (-.f64 y (+.f64 (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 z (-.f64 y x))) (*.f64 a (-.f64 y x))) t) (*.f64 (/.f64 a t) (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t)))): 0 points increase in error, 0 points decrease in error
      (-.f64 y (+.f64 (/.f64 (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x))) t) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x)))) (*.f64 t t))))): 26 points increase in error, 2 points decrease in error
      (-.f64 y (+.f64 (/.f64 (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x))) t) (/.f64 (*.f64 a (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x)))) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 y (neg.f64 (+.f64 (/.f64 (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x))) t) (/.f64 (*.f64 a (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x)))) (pow.f64 t 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (/.f64 (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x))) t) (/.f64 (*.f64 a (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x)))) (pow.f64 t 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 -1 (+.f64 (/.f64 (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 y x) z)) (*.f64 a (-.f64 y x))) t) (/.f64 (*.f64 a (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x)))) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 -1 (+.f64 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t) (/.f64 (*.f64 a (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 z (-.f64 y x))) (*.f64 a (-.f64 y x)))) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 -1 (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (*.f64 a (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x)))) (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 a (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x)))) (pow.f64 t 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 y (*.f64 -1 (/.f64 (*.f64 a (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x)))) (pow.f64 t 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (*.f64 a (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x)))) (pow.f64 t 2))))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> unsub-neg_binary64 (-.f64 y (/.f64 (*.f64 a (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x)))) (pow.f64 t 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 y (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 (*.f64 z (-.f64 y x)) (*.f64 a (-.f64 y x))) a)) (pow.f64 t 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 y (/.f64 (*.f64 (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 y x) z)) (*.f64 a (-.f64 y x))) a) (pow.f64 t 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 y (/.f64 (*.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) a) (Rewrite=> unpow2_binary64 (*.f64 t t)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 y (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t) (/.f64 a t)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 2 points increase in error, 26 points decrease in error
      (+.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 y (*.f64 (neg.f64 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t)) (/.f64 a t)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))) (/.f64 a t))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x)))) t)) (/.f64 a t))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 (/.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x))))) t) (/.f64 a t))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (*.f64 t t)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 26 points increase in error, 2 points decrease in error
      (+.f64 (+.f64 y (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (pow.f64 t 2)) y)) (*.f64 -1 (/.f64 (-.f64 (*.f64 (-.f64 y x) z) (*.f64 a (-.f64 y x))) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (pow.f64 t 2)) y) (*.f64 -1 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 (-.f64 y x) z) t) (/.f64 (*.f64 a (-.f64 y x)) t))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (pow.f64 t 2)) y) (*.f64 -1 (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 z (-.f64 y x))) t) (/.f64 (*.f64 a (-.f64 y x)) t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (pow.f64 t 2)) y) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 (*.f64 z (-.f64 y x)) t)) (*.f64 -1 (/.f64 (*.f64 a (-.f64 y x)) t))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (pow.f64 t 2)) y) (*.f64 -1 (/.f64 (*.f64 z (-.f64 y x)) t))) (*.f64 -1 (/.f64 (*.f64 a (-.f64 y x)) t)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z (-.f64 y x)) t)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 (-.f64 y x) z)) (*.f64 -1 (*.f64 a (-.f64 y x)))) a) (pow.f64 t 2)) y))) (*.f64 -1 (/.f64 (*.f64 a (-.f64 y x)) t))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.063120766029758 \cdot 10^{+157}:\\ \;\;\;\;y - \left(y - x\right) \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t \leq 7.239090729937892 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(\frac{a}{t} + 1\right) \cdot \left(\frac{y - x}{t} \cdot \left(a - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	4688

\[\begin{array}{l} t_1 := y + \left(\frac{a}{t} + 1\right) \cdot \left(\frac{y - x}{t} \cdot \left(a - z\right)\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{-231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	7.9
Cost	4432

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + \frac{a - z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 3
Error	20.3
Cost	1232

\[\begin{array}{l} t_1 := y - \frac{y - x}{\frac{t}{z - a}}\\ \mathbf{if}\;t \leq -2.3613292154529745 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	20.5
Cost	1232

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3613292154529745 \cdot 10^{+145}:\\ \;\;\;\;y - \frac{y - x}{\frac{t}{z - a}}\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{a - z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 5
Error	20.3
Cost	1232

\[\begin{array}{l} t_1 := y - \left(y - x\right) \cdot \frac{z - a}{t}\\ \mathbf{if}\;t \leq -2.3613292154529745 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{a - z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 6
Error	39.1
Cost	1176

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6019423908491996 \cdot 10^{+156}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 7.239090729937892 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 8.9690232009782 \cdot 10^{+168}:\\ \;\;\;\;a \cdot \left(-\frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Error	22.0
Cost	1168

\[\begin{array}{l} t_1 := y + x \cdot \frac{z - a}{t}\\ \mathbf{if}\;t \leq -5.315115461530307 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{a - z}{\frac{-t}{x}}\\ \end{array} \]

Alternative 8
Error	25.0
Cost	1104

\[\begin{array}{l} t_1 := y + x \cdot \frac{z - a}{t}\\ \mathbf{if}\;t \leq -5.315115461530307 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	22.0
Cost	1104

\[\begin{array}{l} t_1 := x + z \cdot \frac{y - x}{a}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 10^{-25}:\\ \;\;\;\;y - z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;a \leq 1.4383997814715265 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.5306033505464576 \cdot 10^{+114}:\\ \;\;\;\;y - x \cdot \frac{a}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	21.8
Cost	1104

\[\begin{array}{l} t_1 := y + x \cdot \frac{z - a}{t}\\ \mathbf{if}\;t \leq -5.315115461530307 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	35.2
Cost	976

\[\begin{array}{l} t_1 := y - x \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -2.6019423908491996 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.87094968082562 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	35.2
Cost	976

\[\begin{array}{l} t_1 := y - x \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -2.6019423908491996 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.87094968082562 \cdot 10^{+26}:\\ \;\;\;\;x + \frac{t \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	29.1
Cost	976

\[\begin{array}{l} t_1 := x + z \cdot \frac{y}{a}\\ t_2 := y - x \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -5.315115461530307 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	28.9
Cost	976

\[\begin{array}{l} t_1 := y - x \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -5.315115461530307 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.962723698611694 \cdot 10^{+40}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	24.4
Cost	976

\[\begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-33}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{-28}:\\ \;\;\;\;y - z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;a \leq 1.4383997814715265 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{x}{\frac{a}{z}}\\ \mathbf{elif}\;a \leq 3.5306033505464576 \cdot 10^{+114}:\\ \;\;\;\;y - x \cdot \frac{a}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 16
Error	38.6
Cost	848

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6019423908491996 \cdot 10^{+156}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -4828750041198972000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.701454154220026 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.6741949626916594 \cdot 10^{+170}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 17
Error	31.8
Cost	712

\[\begin{array}{l} t_1 := y - x \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -5.315115461530307 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.239090729937892 \cdot 10^{+72}:\\ \;\;\;\;x - \frac{x}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	36.7
Cost	592

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{-28}:\\ \;\;\;\;y\\ \mathbf{elif}\;a \leq 1.4383997814715265 \cdot 10^{+53}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.5306033505464576 \cdot 10^{+114}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 19
Error	45.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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