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

Average Error: 24.5 → 7.2

Time: 43.2s

Precision: binary64

Cost: 10704

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ t_2 := x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-205}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-281}:\\ \;\;\;\;t + \left(\frac{a}{z} + 1\right) \cdot \frac{t - x}{\frac{z}{\mathsf{fma}\left(-1, y, a\right)}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- x t) (- z a)) x))
        (t_2 (- x (/ (* (- t x) (- z y)) (- a z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-205)
       t_2
       (if (<= t_2 2e-281)
         (+ t (* (+ (/ a z) 1.0) (/ (- t x) (/ z (fma -1.0 y a)))))
         (if (<= t_2 5e+286) t_2 t_1))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((x - t) / (z - a)), x);
	double t_2 = x - (((t - x) * (z - y)) / (a - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-205) {
		tmp = t_2;
	} else if (t_2 <= 2e-281) {
		tmp = t + (((a / z) + 1.0) * ((t - x) / (z / fma(-1.0, y, a))));
	} else if (t_2 <= 5e+286) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(x - t) / Float64(z - a)), x)
	t_2 = Float64(x - Float64(Float64(Float64(t - x) * Float64(z - y)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-205)
		tmp = t_2;
	elseif (t_2 <= 2e-281)
		tmp = Float64(t + Float64(Float64(Float64(a / z) + 1.0) * Float64(Float64(t - x) / Float64(z / fma(-1.0, y, a)))));
	elseif (t_2 <= 5e+286)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(t - x), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-205], t$95$2, If[LessEqual[t$95$2, 2e-281], N[(t + N[(N[(N[(a / z), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(z / N[(-1.0 * y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+286], t$95$2, t$95$1]]]]]]

Target

Original	24.5
Target	12.2
Herbie	7.2

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0 or 5.0000000000000004e286 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 62.0

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

   2. Simplified 17.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)} \]
      Proof
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 x t) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 x t)))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 x t)))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 x) t))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 x)) t)) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 x)))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 t x))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (neg.f64 (-.f64 t x)) (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (neg.f64 (-.f64 t x)) (-.f64 z a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 (-.f64 t x))) (*.f64 -1 (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 (-.f64 t x)))) (*.f64 -1 (-.f64 z a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (Rewrite=> remove-double-neg_binary64 (-.f64 t x)) (*.f64 -1 (-.f64 z a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= +-commutative_binary64 (+.f64 a (neg.f64 z)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= sub-neg_binary64 (-.f64 a z))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))) x)): 19 points increase in error, 24 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) x): 100 points increase in error, 27 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1e-205 or 2e-281 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 5.0000000000000004e286

   1. Initial program 1.7

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

   if -1e-205 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 2e-281

   1. Initial program 49.4

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

   2. Taylor expanded in z around inf 13.0

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

   3. Simplified 10.1

      \[\leadsto \color{blue}{t + \left(\frac{a}{z} + 1\right) \cdot \frac{t - x}{\frac{z}{\mathsf{fma}\left(-1, y, a\right)}}} \]
      Proof
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (-.f64 t x) (/.f64 z (fma.f64 -1 y a))))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (-.f64 t x) (/.f64 z (fma.f64 -1 y (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 a)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (-.f64 t x) (/.f64 z (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 -1 y) (neg.f64 a))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t x) (-.f64 (*.f64 -1 y) (neg.f64 a))) z)))): 40 points increase in error, 17 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (*.f64 -1 y) (-.f64 t x)) (*.f64 (neg.f64 a) (-.f64 t x)))) z))): 2 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 y (-.f64 t x)))) (*.f64 (neg.f64 a) (-.f64 t x))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 a (-.f64 t x))))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 a (-.f64 t x))))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (/.f64 (Rewrite=> distribute-lft-out--_binary64 (*.f64 -1 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 (+.f64 (/.f64 a z) 1) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (/.f64 a z) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))))): 2 points increase in error, 3 points decrease in error
      (+.f64 t (+.f64 (*.f64 (/.f64 a z) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (+.f64 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 a z) (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (+.f64 (neg.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) (*.f64 z z)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 39 points increase in error, 3 points decrease in error
      (+.f64 t (+.f64 (neg.f64 (/.f64 (*.f64 a (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) (Rewrite<= unpow2_binary64 (pow.f64 z 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 a (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) (pow.f64 z 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 t (*.f64 -1 (/.f64 (*.f64 a (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) (pow.f64 z 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (*.f64 a (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))))) (pow.f64 z 2)))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (/.f64 (*.f64 -1 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) a))) (pow.f64 z 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 -1 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) a)) (pow.f64 z 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (/.f64 (*.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (*.f64 -1 (*.f64 a (-.f64 t x))))) a) (pow.f64 z 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (*.f64 -1 (*.f64 a (-.f64 t x)))))) (pow.f64 z 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (/.f64 (*.f64 a (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (*.f64 -1 (*.f64 a (-.f64 t x))))) (pow.f64 z 2))) (*.f64 -1 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 y (-.f64 t x)) z) (/.f64 (*.f64 a (-.f64 t x)) z))))): 1 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 t (/.f64 (*.f64 a (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (*.f64 -1 (*.f64 a (-.f64 t x))))) (pow.f64 z 2))) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z)) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 t (/.f64 (*.f64 a (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (*.f64 -1 (*.f64 a (-.f64 t x))))) (pow.f64 z 2))) (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z))) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z)) (+.f64 t (/.f64 (*.f64 a (-.f64 (*.f64 -1 (*.f64 y (-.f64 t x))) (*.f64 -1 (*.f64 a (-.f64 t x))))) (pow.f64 z 2))))) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ \mathbf{elif}\;x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z} \leq -1 \cdot 10^{-205}:\\ \;\;\;\;x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z}\\ \mathbf{elif}\;x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z} \leq 2 \cdot 10^{-281}:\\ \;\;\;\;t + \left(\frac{a}{z} + 1\right) \cdot \frac{t - x}{\frac{z}{\mathsf{fma}\left(-1, y, a\right)}}\\ \mathbf{elif}\;x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	10704

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ t_2 := x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot a - y \cdot \left(t - x\right)}{z}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	8.4
Cost	4432

\[\begin{array}{l} t_1 := t + \frac{a - y}{\frac{z}{t - x}}\\ t_2 := x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-205}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-281}:\\ \;\;\;\;t - \left(y - a\right) \cdot \frac{t - x}{z}\\ \mathbf{elif}\;t_2 \leq 10^{+304}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.1
Cost	4432

\[\begin{array}{l} t_1 := t + \frac{a - y}{\frac{z}{t - x}}\\ t_2 := x - \frac{\left(t - x\right) \cdot \left(z - y\right)}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot a - y \cdot \left(t - x\right)}{z}\\ \mathbf{elif}\;t_2 \leq 10^{+304}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	23.3
Cost	1368

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t}{a}\\ t_2 := t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2.9426793673236367 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-223}:\\ \;\;\;\;x - y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-291}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-247}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	20.1
Cost	1232

\[\begin{array}{l} t_1 := t + \frac{a - y}{\frac{z}{t - x}}\\ \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 10^{-100}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 5.738369557299789 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.6
Cost	1232

\[\begin{array}{l} \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-155}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 3.5283904728166974 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{x}{\frac{z - a}{y - z}}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{a - y}{\frac{z}{t - x}}\\ \end{array} \]

Alternative 7
Error	19.5
Cost	1232

\[\begin{array}{l} \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-155}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 3.5283904728166974 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{x}{\frac{z - a}{y - z}}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t - \left(y - a\right) \cdot \frac{t - x}{z}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	1232

\[\begin{array}{l} t_1 := x + \frac{y - z}{\frac{a - z}{t}}\\ \mathbf{if}\;z \leq -1.2023703340738331 \cdot 10^{+119}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-289}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 2.085151857107415 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t - \left(y - a\right) \cdot \frac{t - x}{z}\\ \end{array} \]

Alternative 9
Error	22.8
Cost	1104

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t}{a}\\ t_2 := t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2.9426793673236367 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-223}:\\ \;\;\;\;x - y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	20.6
Cost	1104

\[\begin{array}{l} t_1 := t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 5.738369557299789 \cdot 10^{+19}:\\ \;\;\;\;t \cdot \frac{z - y}{z - a}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	34.9
Cost	844

\[\begin{array}{l} t_1 := t - \frac{t}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -5.149600259341313 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	32.5
Cost	844

\[\begin{array}{l} t_1 := t - \frac{t}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -5.149600259341313 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	32.7
Cost	844

\[\begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -0.003637529066001391:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	33.1
Cost	844

\[\begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -0.003637529066001391:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x - y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	22.8
Cost	840

\[\begin{array}{l} t_1 := t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	20.8
Cost	840

\[\begin{array}{l} t_1 := t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	19.9
Cost	840

\[\begin{array}{l} t_1 := t + \left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	36.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -0.003637529066001391:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3461495599211315 \cdot 10^{+138}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 19
Error	27.7
Cost	712

\[\begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -700840357537700.9:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	35.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3461495599211315 \cdot 10^{+138}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 21
Error	35.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;z \leq -1.88177966163057 \cdot 10^{+61}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.4805273746873535 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 22
Error	45.8
Cost	64

\[t \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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